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\hyphenation{mi-ni-mi-zing  mi-ni-mi-zers quadruple dif-fe-ren-tial}
\begin{document}
\title{Sub-Riemannian  Metrics: Minimality of  Abnormal  Geodesics versus Subanalyticity}
\author{Andrei  A. Agrachev$\footnotesize ^{1}$ \and
Andrei V. Sarychev$\footnotesize ^{2}$}
\date{}
\maketitle
\setcounter{footnote}{1}
\footnotetext{Leading Research Associate,
Steklov Mathematical Institute, Russian Academy of Sciences,
Gubkina ul. 8, 117966, Moscow, Russia; e-mail: andrei@agrachev.mian.su}
\addtocounter{footnote}{1}
\footnotetext{Invited Associate Professor,
Departamento de Matem\'{a}tica,
Universidade de Aveiro, 3810, Aveiro, Portugal;
e-mail: ansar@mat.ua.pt }


\section{\small\bf Introduction}
\label{intr}
\setcounter{equation}{0}

\hspace*{5mm} We study  local properties of {\em real-analytic}  sub-Riemannian metrics.

Recall that a
{\em sub-Riemannian  structure} on a connected $n$-dimensional Riemannian
manifold $M$ is  defined by means of  a   {\em distribution}  $\cal D$ on $M$. Recall that  distribution is a subbundle of $TM$.  Below we assume $\cal D$ to be  {\em bracket generating}.
This means that for  each point $q  \in M$  there exists
integer $\kappa_q \geq 1$ such that the values of  the iterated Lie brackets of orders $\leq \kappa_q$
of the vector fields subject  to $\cal D$ span the  tangent space $T_{q}M$.
The minimal integer $\kappa_{q} \geq 1$ is called {\em degree of nonholonomy}  of the distribution $\cD$ at the point $q$.


{\em Admissible } or {\em horizontal paths}
are Lipschitzian curves
$t \mapsto q(t)$     which
are  tangent to the distribution at (almost)  each
of their  points: $\dot{q}(t) \in {\cal D}_{q(t)}$.
As far as $\cD$ is bracket generating
then according to Rashevsky-Chow theorem
(see subsection~\ref{distr})    any two points
$q^{0},q^{1} \in M$  can be connected by a Lipschitzian   admissible  path  $q(t)$.

According to classical existence results of optimal control theory (Filippov theorem; see \cite{Fi}) for any pair of points from a sufficiently small neighborhood of $q^{0}$
there  exists  (possibly non unique)  path  of  minimal  length
in the set of admissible paths connec\-ting these two  points.
Global existence can be asserted under standard assumption of completeness.

Attributing to each
couple of points the infimum of the lengths  of admissible paths  we obtain so-called
{\em sub-Riemannian} or {\em Carnot-Caratheodory metric}
on the mani\-fold $M$.
The properties of the sub-Riemannian metrics and of  corresponding
length minimizers ({\em sub-Riemannian geodesics})
were intensively studied during the last decade not only for their  own
sake but also due to numerous applications  to  physics.



There are many similarities between  Riemannian and  sub-Riemannian metrics, and  also some major differences. One of these  differences    is that the 'exponential maps' in the sub-Riemannian case have singularity  at the origin
$q^{0}$, as far as  all geodesics are  necessarily tangent to a proper  subspace
${\cal D}_{q^{0}} \subset T_{q^{0}}M$.
The conjugate locus
of a point $q^{0} \in M$ in the
sub-Riemannian case contains $q^{0}$ in its closure. Due to it there are arbitrarily short geodesics paths which  fail  to be  length-minimizers. This implies non smoothness of small sub-Riemannian spheres on the contrary to the case of small Riemannian spheres which are $C^{\omega}$-diffeomorphic to  Euclidean spheres.


In the sub-Riemannian case the set of admissible paths,  connecting two given points,   may fail  to be a Banach manifold  (on the contrary to the Riemannian case).
It may have singularities which correspond to
an interesting phenomenon discovered  some years ago  -
an occurrence of so-called  {\em abnormal sub-Riemannian length-minimizers} or
{\em  length minimizing abnormal sub-Riemannian geodesics}. The first
example has been constructed  by R.Montgomery (\cite{Mon}), further examples,
theoretical study of this phenomenon  and bibliographic references can be found in \cite{AS1,AS2,AS3, Mon2,SuC,SL}.

The abnormal sub-Riemannian geodesics
do not depend on Riemannian metric and  are determined by
the distribution $\cal D$. As it has been just said  they  are singular 'points'  in  the set of admissible paths
on the manifold connecting  two given points.
They may exhibit some  exotic properties, like for example be
{\em rigid}, i.e.
isolated  in $W_{1,\infty}$-metric of Lipschitzian paths,  in the above mentioned set. For an  abnormal sub-Riemannian geodesic
the adjoint covector
in the Hamiltonian  form of  geodesic equation is   orthogonal to the distribution
$\cal D$ along the geodesic path.

It is clear that if one studies properties of
sub-Riemannian metrics then  some pathologies  arise
exactly  when  abnormal sub-Riemannian minimizers appear.
On the contrary if there are no abnormal length-minimizers
one can expect 'more  regu\-la\-ri\-ty'  for  the corresponding  metrics.

To formulate more rigorously what we mean by   regularity we have to recall
that a sub-Riemannian $\rho$-sphere - the set of points located at
(Carnot-Caratheodory) distance $\rho >0$ from the given point $q^{0}$ -
is a subset  of the   image of the {\em sub-Riemannian exponential map}.
(The whole image - the set of end-points of sub-Riemannian geodesics of
given length $\rho$ -  is called {\em wave front}.) This  map is defined
on a subset of $T^{*}M$ and is real analytic.  Therefore the 'right'
category of sets to which the sub-Riemannian spheres may belong is 
the one of {\em subanalytic sets}
(see \cite{Ga,Ha,Hi}), which are exactly
the images of semianalytic sets under {\em proper}
real analytic maps.
The subanalytic sets form a wider class than the semianalytic ones, but still share with them an important regularity property -
they  admit {\em stratifications}
(\cite{Ha,Hi}).

The subanalytic sets have been utilized in control theory
since the end of 70's in order to  construct  optimal synthesis and to study  the structure
of reachable sets of control systems (\cite{Bru,Su7}). It has been established  that even this large category is not enough to describe all control-theoretic objects. Thus H.J.Sussmann and S.Lojasiewicz Jr. constructed (\cite{LS}) examples of control systems with non subanalytic reachable sets.

The sub-Riemannian spheres a priori could be expected to exhibit more regularity as far as they are parts of  reachable sets of {\em symmetric control systems}.
They indeed are more regular when
abnormal length-minimizers
are lacking.

A good example of this kind is provided by  a   class  of  {\em fat} distributions.
A distribution is called fat (\cite{BH})
at a point $q \in M$ if for
any vector field $X$ with
$X(q) \in {\cal D}_{q}, \ X(q) \neq 0,$ there holds
 $$[X, {\cal D}](q)={\cal D}(q)+\span\{[X,Y](q): \ Y \in {\cal D}\}=T_{q}M.$$
 It is easy  to see that for a distribution which is fat
at the point $q^{0} \in M$, there are
no abnormal sub-Riemannian geodesic paths  starting   at $q^{0}$.
H.J.Sussmann (\cite{Sus}) and Zhong Ge (\cite{ZG})   studying the   properties of sub-Riemannian fat metrics have  established their  subanalyticity.

On the other hand it is known that occurrence of abnormal length-minimizers may destroy  the regularity.  For example sub-Riemannian metric  arising from {\em Martinet distribution} -     first example in which abnormal length-minimizers have been discovered - fails  to be subanalytic, as has been established by A.Agrachev, B.Bonnard, M.Chyba, I.Kupka (\cite{ABCK}).

In the present paper we make the next  step studying some classes of sub-Riemannian structures which do possess abnormal geodesics, but these latter are not length-minimizing.
For some of these classes we manage to prove   that the corresponding sub-Riemannian metrics are subanalytic.





Before introducing these classes of distributions  let us recall that
earlier in  \cite{AS2}
we  derived  {\em necessary}  conditions for  rigidity and weak
(=$W_{1,\infty}$-local)  minimality of
abnormal sub-Riemannian geodesics.
These conditions are counterparts of
optimality conditions for singular extremals
known in optimal control theory since 60's. One of them  called  {\em Goh condition},
appeared in \cite{AS2}     as  necessary condition  for rigidity
%% (= isolatedness in $W_{1,\infty}$-metric)
of abnormal sub-Riemannian geodesics
(see \cite[Proposition  4.3]{AS2}).
For the sub-Riemannian length-minimization problems the Goh condition  amounts to
orthogonality  (along the abnormal geodesic)  of the Hamiltonian multiplier
 to the Lie square
${\cal D}^{2}(q)=\left({\cal D}+[{\cal D}, {\cal D}]\right)(q)$
of the distribution.
In \cite{AS2}  it has been also stated   (Proposition 4.2) that
the Goh condition is necessary  for weak minimality
whenever  abnormal geodesic path is {\em strictly
abnormal}, i.e. when {\em all} possible
Hamiltonian multipliers corresponding
to this path annihilate (or are orthogonal to)    the distribution.
In the present paper we will obtain this  necessary condition   on the basis of a more general result
which  relates    Morse index
of a critical point of a smooth mapping to  its local openness  (Theorem~\ref{wmn}).

There are  some classes of distributions  for which
abnormal geodesics do exist but
can not satisfy the Goh condition.
For example, it   happens   when the Lie square
${\cal D}^{2}(q)=({\cal D}+[{\cal D}, {\cal D}])(q)$
of the distribution $\cal D$ at the point $q \in M$ coincides with
the tangent space $T_{q}M$. In this case there may exist strictly abnormal
geodesic paths  starting at $q$ but due to the Goh necessary condition no one of them  is weak length minimizer.
We shall call  (following H.J.Sussmann)
these distributions  {\em 2-generating}   at the point
$q \in M$.
Obviously all fat distributions are 2-generating.

A larger class is formed by
distributions  $\cal D$, which we call  {\em medium fat}.
They satisfy the condition
that for  any vector field $X \in {\cal D}, \ X(q) \neq 0:$
$$[X, {\cal D}^{2}](q)={\cal D}^{2}(q)+\span\{[X,Y](q):
\ Y \in {\cal D}^{2}\}=T_{q}M.$$
The class of medium fat distributions is bigger than the one  of $2$-generating distributions. It is also bigger than the class of distributions with fat Lie square.
 Also for medium fat distributions we prove
that there are
no strictly abnormal length-mi\-ni\-mi\-zers (Theorem~\ref{mefa}).
%%\marginpar{\small\bf proof?}

It turns out that the  same conditions which determine
weak length minimality  or non minimality of  strictly abnormal geodesic paths
 also have crucial influence on the properties of the sub-Riemannian metrics.

The main results obtained in this paper include the following statements:
\begin{itemize}
\item if $\cal D$ is a germ at $q^{0}$ of a 2-generating distribution then the germ of corresponding sub-Riemannian (Carnot-Caratheodory) metric $\rho_{\cal D}(q,q')$ is subanalytic  (Theorem~\ref{2Dsu})
\footnote{This result has been independently obtained by
S.Jacquet in his PhD thesis, Savoye, 1998}
;
\item if $\cal D$ is a germ at $q^{0}$ of a medium fat distribution then the germ of corresponding sub-Riemannian (Carnot-Caratheodory) metric $\rho_{\cal D}(q,q')$ is subanalytic outside of the diagonal $\{q=q'\}$ (Theorem~\ref{mefas}).
\end{itemize}

The proofs of these statements are based on
a study of
second variation along normal geodesics  and
on a {\em compactness} result
for  the set of length minimizers (of given length) for the medium fat and 2-generating  distributions.
  Obviously the second statement  refers to a bigger class of distributions, though there remains a problem of studying the metric at the diagonal.

The following statements  describe how large are these two
classes of distributions:
\begin{itemize}
\item generic germ of rank $r$  distribution on $n$-dimensional manifold
with $r \leq n \leq \frac{r(r+1)}{2}$ is $2$-generating
(Proposition~\ref{getu});
\item generic germ of rank $r$  distribution on $n$-dimensional manifold
with $r \leq n \leq r+(r-1)^{2}$ is medium fat (Theorem~\ref{gem}).
\end{itemize}

The following tables illustrate these results representing
information about   (low) dimensions for which the properties of being   2-generating or  medium fat
are generic (evidently one assumes $\dim M \geq \dim \cD$).


\vspace*{2mm}

{\small\bf 2-generating distributions}

\vspace*{1mm}

\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
$\dim \cD$ & $3$ & $4$ & $5$  & $6$ & $\cdots$ & r  \\
\hline
$\dim M$  & $\leq 6$ & $\leq 10$ & $\leq 15$ & $\leq 21$
& $\cdots$ & $\leq r(r+1)/2$ \\
\hline
\end{tabular}



\vspace*{3mm}

{\small\bf Medium fat distributions}

\vspace*{1mm}


\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
$\dim \cD$ & $3$ & $4$ & $5$  & $6$ & $\cdots$ & r  \\
\hline
$\dim M$  & $\leq 7$ & $\leq 13$ & $\leq 21$ & $\leq 31$
& $\cdots$ & $\leq r+(r-1)^{2}$ \\
\hline
\end{tabular}



%%Thus, in particular, generic germs  of $3$-distributions %%on the manifolds
%%of dimension $\leq 6$, generic germs of $4$-distributions %%on   the manifolds
%%of dimension $\leq 10$,  generic germs of $5$-%%distributions on   the manifolds
%%of dimension $\leq 15$ are $2$-generating, while
%%generic germs of
%%$3$-distributions
%%on the manifolds
%%of dimension $\leq 7$, generic germs of $4$-distributions %%on   the manifolds
%%of dimension $\leq 13$,  generic germs of $5$-%%distributions on   the manifolds
%%of dimension $\leq 21$ are medium fat.

\vspace*{4mm}


The  present paper has the following structure. Section~\ref{pre}
contains some basic
definitions concerning analytic, semianalytic
and subanalytic sets, distributions on manifolds and formulae  of chronological calculus for flows. These formulae    will be   needed for computation of  first and second variations  along a geodesic.
In Section~\ref{srlm} we introduce  the sub-Riemannian
length-minimization problem and formulate the corresponding
first-order necessary minimality condition - Hamiltonian form of geodesic  equation. We also introduce normal and abnormal  sub-Riemannian  geodesics.
In the subsection~\ref{svar} we introduce first and second
variations for the length-minimization problem.
In the Section~\ref{morse} we formulate some necessary conditions of
weak minimality  for sub-Riemannian geodesics
in terms of their Morse index (Theorem~\ref{nemi}) .
These conditions are  derived from a general result
relating Morse index of a critical point of smooth mapping
to stable local openness  of this map at this point
(Theorem~\ref{wmn}). The Goh condition appears then as   necessary condition
for finiteness of Morse index and therefore for weak minimality of strictly abnormal geodesic paths   (Proposition~\ref{gon} and Theorem~\ref{legom}). Theorem~\ref{mefa} affirms
 absence of abnormal length-minimizers for medium fat and 2-generating distributions.
%%In the next section~\ref{coponomi} we introduce notion
%%of conjugate point for corank $k$ normal sub-Riemannian
%%geodesic  for a real analytic distribution and formulate
%%Jacobi-type minimality condition for a corank $k>1$
%%geodesic (for corank 1  geodesic this turns into
%%Jacobi condition).
Section~\ref{ssrs} contains main results of the paper on
subanalyticity: we prove subanalyticity of metrics  for medium fat  sub-Riemannian structures (Theorems~\ref{2Dsu} and \ref{mefas}).
In the Section~\ref{gener} we evaluate the broadness of
the classes  of 2-generating and medium fat distributions  (Proposition~\ref{getu} and Theorem~\ref{gem}).  Section~\ref{proo} contains
major  proofs.





\section{Preliminaries}
\label{pre}
\setcounter{equation}{0}

In what follows the manifold $M$ is supposed to be connected.

\subsection{Analytic, Semianalytic and Subanalytic Sets}
\label{anal}



Recall that a subset $S$ of a real analytic   manifold $M$ is called {\em analytic} if $\forall p \in S$ there exists a neighborhood $U$ of $p$ in $M$ such that
$U \cap S$ is defined by a finite system of equations
$\varphi_{i}(x)=0, i=1, \ldots , m$, where
$\varphi_{i}:M \rightarrow R$ are real analytic functions.

 A subset $S$ of a real analytic  manifold $M$ is called {\em semianalytic} if $\forall p \in S$ there exists a neighborhood $U$ of $p$ in $M$ such that
$U \cap S$ is defined by a finite system of inequalities $\varphi_{i}(x) \geq 0, i=1, \ldots , m, \
\chi_{i}(x) > 0, i=1, \ldots , s$, where
$\varphi_{i}:M \rightarrow  R, \chi_{j}:M \rightarrow  R,$  are real analytic functions.

An  extension of the class of semianalytic sets  has been introduced (\cite{Ga,Hi,Ha})
in order to recover an important property of 'closeness under projection' which is valid for semialgebraic sets (Tarski-Seidenberg theorem)   but  may fail   for the semianalytic ones.  The images of semianalytic sets under   analytic maps  do not in general belong to the semianalytic category.

A  subset $S$ of a real analytic  manifold $M$ is called {\em subanalytic} if  there exists a semianalytic subset $T$ of a  real analytic manifold $N$ and a  real analytic map $f:N \rightarrow M$, which is proper on $\mbox{clos} T$, and such that $f(T)=S$.
Recall that a map $f$ is {\em proper} if the inverse images $f^{-1}(K)$ of the compact sets $K$ are compact.

Proper analytic maps map subanalytic sets into subanalytic ones. The class of subanalytic sets is also closed under the operations of taking finite unions, finite intersections, complements and under inverse images of analytic maps.

The subanalytic sets though being much more general than the semianalytic ones share with them an important regularity property -  they are  {\em stratifiable.}

{\em Stratum} is an analytic connected embedded submanifold
of $M$. {\em Whitney stratification}
  $\cal S$ in $M$ is a set of pairwise disjoint strata, such that:

i) $\cal S$ is locally finite, i.e. for any compact $K \subset M$ only a finite number of elements of $\cal S$ have nonempty intersections with $K$;

ii) for any $S \in {\cal S}$ the {\em frontier}
$\mbox{Fr} S=\mbox{clos}S \setminus S$ is a union of elements of $\cal S$
of dimension strictly less than $\dim S$;

iii) for any pair
of strata $(S_\alpha , S_\beta)$ such that
$S_\alpha \subset \mbox{clos}S_\beta$, the following {\em Whitney
conditions} hold:

suppose that sequences $x_i \in S_\beta$ and $y_i \in S_\alpha$
converge to one and the same point $y \in S_\alpha$,
and (in some local coordinates on $M$) the secants $l_i=\overline{x_i y_i}$
converge to a straight line $l$ and that the tangent spaces
$T_{x_i}S_\beta$ converge to a subspace $\cal T$ then
iv) $T_y S_\alpha \subset {\cal T}$ and v) $l \subset \cal T$.

The Whitney stratifications enjoy some nice properties; in particular, they
admit triangulations.


$\cal S$ is a
{\em stratification of} $A$, if
$\cup_{S \in {\cal S}}=A$.

The following fact is fundamental for the subanalytic subsets:
{\em any closed subanalytic subset admits Whitney stratification};
see \cite{GM} for details and references.


A map  $f:M \rightarrow N$ between two $C^{\omega}$-manifolds is called {\em subanalytic} if its graph is a subanalytic subset of $M \times N$.


\subsection{Distributions and sub-Riemannian metrics}
\label{distr}

A {\em distribution} $\cal D$ on $M$
is a subbundle of tangent bundle ${\cal T}M$.  A  vector field $X$ is
subject to $\cal D$ if $X(q) \in {\cal D}_{q} \subset {\cal T}_{q}M$
for every $q \in M$.



Generalizations of distributions are {\em differential systems}
or {\em distributions with singularities}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{footnote}{0}
\footnote{Not to be mixed with the differential systems determined
by the differential forms; those have different kind of singularities}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for which $\dim \cD_{q}$ may vary with $q \in M$.
A differential system can be viewed as  $C^{\infty}(M)$- submodule
of $\mbox{Vect}M;$
then distributions correspond to projective
$C^{\infty}-$mo\-dules. Locally one may treat germ of  distribution
as a free module.

If $\cD$ is a germ of  differential system, then taking $C^{\infty}-$modules
generated by the Lie brackets of order $\leq k, \ k=1, \ldots , \  $
of the vector fields,  subject  to $\cD$,
one obtains an expanding sequence
of differential systems:
\[ \cD \subseteq \cD^{2}=[\cD ,\cD] \cdots
\subseteq \cD^{k}=[\cD,\cD^{k-1}] \subseteq \cdots . \]
For any $q \in M$ the sequence of subspaces
\[ \cD_{q} \subseteq \cdots \subseteq \cD^{k}_{q} \subseteq {\cal T}_{q}M \]
is called {\em flag of the differential system} $\cD$ at the point
$q \in M$, while the sequence
$n_{1}(q) \leq \cdots  \leq n_{k}(q) \leq \cdots $,
where $n_{i}(q)= \dim \cD^{i}_{q}$,
is called {\em vector of growth of the differential system $\cD$
at the point} $q$.
Differential system is called {\em completely nonholonomic}
or {\em having complete  Lie rank} or {\em bracket generating} at a point $q \in M$ if
$\cD_{q}^{\kappa}={\cal T}_{q}M$ for some $\kappa$.
This $\kappa$ is called {\em degree of nonholonomy
of the distribution} at $q$.


A locally Lipschitzian path
$\tau \mapsto  q(\tau)$ on $M$ is called {\em admissible} if its tangents
belong to  $\cal D$ for almost all $\tau \in [0,T].$


 A fundamental property of the bracket generating differential systems %
   is established by the following theorem. %
   \newtheorem{RC}{Theorem}[subsection] %
   \begin{RC}[Rashevsky-Chow Theorem; see \cite{Ra,Cho}]         \label{Ra} %
   If a differential system  is bra\-cket generating  %
   on the manifold $M$, then any two points of $M$ %
   can be connected by an admis\-sible path. $\Box$ %
   \end{RC} %



If one deals with distributions ($\dim {\cal D}_{q}= \mbox{const}$) then from existence results of optimal control theory (Filippov theorem) it follows (see \cite{VG})  that, locally,
for any pair of points from a sufficiently small neighborhood of $q^{0}$
there  exists  (possibly non unique)  path  of  minimal  length
in the set of admissible paths connecting these two  points.
  Attributing to each
couple of points the infimum of the lengths of
admissible paths  we obtain so-called
{\em sub-Riemannian} or {\em Carnot-Caratheodory metric}
on the mani\-fold $M$.



One can define in an obvious way a sub-Riemannian $\rho$-sphere
and sub-Riemannian $\rho$-ball centered
at $q^{0} \in M$ as a set of points located at sub-Riemannian distance $\rho$ (respectively $\leq \rho$)
 from $q^{0}$.

The following parallelepiped-like (see \cite{HH,VG})
lower, or interior,  estimate  is valid  for the sub-Riemannian balls.
Let $\kappa >1$ be the degree of nonholonomy
of $\cD$ at  $q^{0}$ and
$n_{1}(q^{0}) \leq \cdots \leq n_{\kappa}(q^{0})$
its vector of growth at the same point.
Let us define  for $i=1, \ldots , n, \
w_{i}=j$, if $n_{j-1} < i \leq n_{j}$.
Consider the  parallelepiped $\Pi_\rho$ in $ R^{n}: \
\Pi_{c,\rho}=\{(x_{1}, \ldots , x_{n}) | \ |x_{i}| \leq c\rho^{w_{i}}\}$.
The following  statement is a corollary of the results of \cite{AGS,Ger,HH}:  if $B_{\rho}$ is sub-Riemannian $\rho$-ball centered at $q^{0}$, then there exists local coordinate system and constants $\rho_{0}>0, c,C>0$
such that for all $\rho < \rho_{0}$:
$ \Pi_{c,\rho} \subset B_{\rho} \subset \Pi_{C,\rho}$.
An easy corollary of this estimate is the fact that
for some constant $c'>0$ and for all  $\rho < \rho_{0}$
the sub-Riemannian balls $B_\rho$ contain
the Riemannian balls of the radius $c'\rho^{\kappa}$
- Riemannian distance from $q^{0}$ a
a sub-Riemannian $\rho$-sphere is estimated
from below by
 $c'\rho^{\kappa}$.



Let us note that even if  $\cD$ is a  distribution
($n_{1}(q) \equiv \mbox{const}$), then still
$\cD^{k}$ may have non constant dimensions -
the growth vector of a distribution in general
changes with $q$.
A germ of distribution is called {\em regular}
if its growth vector  is  constant.
%%For sub-Riemannian balls corresponding to
%%regular distributions a similar
%%upper, or exterior,
%%estimate $ \Pi_{c,\rho} \supseteq B_{\rho}$   is valid  %%(\cite{Ger}).
For regular sub-Riemannian metrics associated with regular distributions    one can derive from the above mentioned estimates
formula for {\em Hausdorff dimension} of the sub-Riemannian manifold:
$$ \dim_{H}M=n_{1}+\sum_{i=2}^{\kappa}i(n_{i}-n_{i-1}).$$



Let us introduce some notions related to the growth vector of
a germ of a distribution.

A distribution is called {\em fat} (\cite{BH}),  if $\forall X \in \cD$:
$$ \ X(q) \neq 0  \Rightarrow [X, {\cal D}](q)={\cal D}(q)+\span\{[X,Y](q):
\ Y \in {\cal D}\}=T_{q}M.$$

A germ of distribution is called
{\em 2-generating} at a point $q \in M$  if  its degree  of
nonholonomy $\kappa \leq 2$,
i.e. $\cD_{q}^{2}={\cal T}_{q}M$.

 We call distribution
{\em medium fat} at $q$, if  $\forall X \in \cD$:
$$[X, {\cal D}^{2}](q)={\cal D}^{2}(q)+\span\{[X,Y](q):
\ Y \in {\cal D}^{2}\}=T_{q}M.$$
The degree of nonholonomy of a medium fat distributions is $\leq 3$.

 Obviously this  latter   property
is weaker than fatness  and even weaker than the fatness of the Lie square ${\cal D}^{2}$.
The class of fat distributions is contained in the class of
2-generating distributions which is contained in the class of
distributions with fat Lie square which is contained in the class of medium fat distributions.


\subsection{Exponential representation of flows: some formulae}
\label{chr}

Here we introduce some notation and  formulae of {\em chronological calculus}  which will be utilized in the next section to obtain expressions for the first and the second variations along a geodesic. The chronological calculus has been developed in \cite{AG2},
one can find  in \cite{AGS} exposition of some  part
 of it which is  sufficient  for our objectives.


A {\em flow}  on $M$ is an absolutely continuous
(with respect to   $\tau \in  R$)
curve $\tau \longrightarrow P_{\tau}$ in the group of
diffeomorphisms Diff~$M$, satisfying the condition
$P_{0}=I$  - the identity diffeomorphism.
We assume all time-dependent vector fields $X_{\tau}$ to be
locally integrable with respect to $\tau$, below they turn out  to be real analytic.
A time-dependent vector field
$X_{\tau}$ defines an ordinary differential equation
$\dot{q}=X_{\tau}(q(\tau)), q(0)=q^{0}$ on the manifold $M$.
If the solutions of this differential equation exist for all
$q^{0} \in M, \tau \in R$, then the vector field $X_{\tau}$
is called {\em complete} and defines a flow on $M$, which is the
unique solution of the (operator) differential equation:
\begin{equation}                                      \label{de}
dP_{\tau}/d \tau =P_{\tau} \circ X_{\tau}, P_{0}=I.
\end{equation}
This solution will be denoted by
$P_{t}=\chro_{0}^{t}X_{\tau}d
\tau$, and is called (see \cite{AG2} or \cite{AGS}) a {\em right chronological
exponential} of $X_{\tau}$.
If the vector field $X_{\tau} \equiv X$ is
time-independent, then the corresponding
flow is denoted by $P_{t}=e^{tX}$.

The chronological exponential admits Volterra expansion   (see \cite{AG2,AGS});
%%\[ \chro_{0}^{t} X_{\tau} d \tau
%%\asymp I+ \sum_{i=1}^{\infty} \int_{0}^{t}
%%d \tau_{1} \int_{0}^{\tau_{1}} d \tau_{2} \ldots
%%\int_{0}^{\tau_{i-1}} d \tau_{i}
%%(X_{\tau_{i}} \circ \cdots X_{\tau_{1})}.   \]
we will need only its  terms of zero-, first- and
second-order:
\begin{equation}
\chro_{0}^{t} X_{\tau} d \tau              \label{P2}
\asymp I+\int_{0}^{t}X_{\tau}d \tau+
\int_{0}^{t}d \tau_{1} \int_{0}^{\tau_{1}}d \tau_{2}
(X_{\tau_{2}} \circ X_{\tau_{1}})+ \cdots .
\end{equation}
%%For time-independent $X$ one obtains
%%\begin{equation}
%%$e^{tX} \asymp I+tX+(t^{2}/2) X \circ X+ \cdots $.
%%\end{equation}

One more tool from the chronological calculus is a 'generalized
variational formula'(see \cite{AG2,AGS} for its drawing).
In this formula we use the notation $\Ad P$  (where $P$ is a diffeomorphism) for the
following inner automorphism of the Lie algebra of vector fields on $M$:
$\Ad PX=P \circ X \circ P^{-1}=P_{*}^{-1}X$. The last
notation stands for the result of translation of the vector
field $X$ by the
differential of the diffeomorphism $P^{-1}$.

The generalized variational formula can be written as
\begin{equation}
\chro_{0}^{t}                         \label{P4}
(\hat{X}_{\tau}+X_{\tau})d \tau=
\chro_{0}^{t}
{\rm Ad}(\chro_{0}^{\tau}
\hat{X}_{\theta}d \theta) X_{\tau}d \tau  \circ
\chro_{0}^{t}\hat{X}_{\tau}d \tau .
\end{equation}

Applying the operator
$\Ad (\chro_{0}^{\tau}\hat{X}_{\theta}d \theta)$
to a vector field $Y$
and differentiating
$$\Ad (\chro_{0}^{\tau}\hat{X}_{\theta}d \theta)Y=
(\chro_{0}^{\tau}\hat{X}_{\theta}d \theta) \circ Y
\circ (\chro_{0}^{\tau}\hat{X}_{\theta}d \theta)^{-1}$$
with respect to  $\tau$
one comes to the equality (see \cite{AG2,AGS}):
\begin{equation}                                          \label{P66}
\frac{d}{d\tau}\Ad \left(\chro_{0}^{\tau}\hat{X}_{\theta}d \theta \right) Y=
\Ad (\chro_{0}^{\tau}\hat{X}_{\theta}d \theta) \circ \ad \hat{X}_{\tau}Y,
\end{equation}
which is of the same form as (\ref{de}). Therefore
$\Ad \left(\chro_{0}^{\tau}\hat{X}_{\theta}d \theta\right)$
can be represented as an
operator chronological exponential
$\chro_{0}^{t} \ad \hat{X}_{\theta}d \theta$.
For a time-independent vector field
$\hat{X}_{\tau} \equiv \hat{X}$ it can be written as
$e^{t \ad \hat{X}}.$
These  exponentials also admit Volterra expansions:
\begin{equation}                               \label{P22}
\chro_{0}^{t} \ad X_{\tau} d \tau
\asymp  I+\int_{0}^{t} \ad X_{\tau}d \tau+
\int_{0}^{t}d \tau_{1} \int_{0}^{\tau_{1}}d \tau_{2}
(\ad X_{\tau_{2}} \circ \ad X_{\tau_{1}})+ \cdots .
\end{equation}
%%and
%%$$e^{t \ad X} \asymp I+t \ad X+(t^{2}/2) \ad X \circ \ad %%X+ \cdots .$$

In this new notation the generalized variational formula
(\ref{P4}) can be represented as:
\begin{equation}                             \label{P34}
\chro_{0}^{t}
(\hat{X}_{\tau}+X_{\tau})d \tau = \chro_{0}^{t}
(\chro_{0}^{\tau}
\ad \hat{X}_{\theta}d \theta) X_{\tau}d \tau
\circ \chro_{0}^{t}\hat{X}_{\tau}d \tau .
\end{equation}





\section{Sub-Riemannian Problem of Length Minimization}
\label{srlm}
\setcounter{equation}{0}

\subsection{Normal and Abnormal Sub-Riemannian  Geodesics.
Weak Minimality and Rigidity}
\label{ncag}

%%In \cite{AS2} we derived a {\em necessary condition %%for
%% rigidity  and/or  weak minimality of abnormal %%geodesic %%paths}, whose counterpart in optimal %%control is called %%{\em Goh condition}.
In this subsection we
  set   a problem of sub-Riemannian length minimization
 (minimization of the Carnot-Caratheodory length), define sub-Riemannian geodesics as  extremals of this minimization problem,  introduce normal and
 abnormal geodesics and define rigidity.

From now on we deal with a real-analytic
{\em sub-Riemannian structure} on
a connected Riemannian manifold $M$.  This structure is defined by means of bracket generating distribution
$\cal D$ on $M$.  Given two points $q^{0}$ and $q^{1}$ we set a problem of finding
admissible path of minimal length connecting
$q^{0}$ with $q^{1}.$  According to Rashevsky-Chow  theorem the set of admissible paths connecting these two points is nonempty.

The problem we set is a generalization of the classical Riemannian length minimization problem,
but in fact there are  essential differences.
Thus the  space of
{\em all} locally Lipschitzian paths, which
connect $q^{0}$ and $q^{1}$, has natural structure of Banach manifold.
Critical points of the length functional on this manifold
are Riemannian geodesics and all paths of minimal length are among them.
On the contrary the space of {\em admissible} paths,
which connect $q^{0}$ and $q^{1}$,  is not, in general, a manifold; it may have
singularities.
These singularities correspond to  {\em abnormal}
sub-Riemannian geodesics, which do not
depend on Riemannian structure on $M$ and are
determined by the distribution $\cal D$.

The term 'abnormal'
came  from optimization theory,
since the  problem of finding minimal admissible
path can be
 formulated as
a Lagrange problem of the Calculus of Variations.
The {\em extremals} of this  latter  problem are called
{\em sub-Riemannian geodesics} and, in particular,
{\em abnormal extremals,} with vanishing Lagrange multiplier for the
(length) functional, are called
{\em abnormal sub-Riemannian geodesics}.

In spite of a lot of activity tended to elimination of
abnormal sub-Riemannian geodesics from the candidates to minimizers, they can be minimal.
Preprint \cite{Mon} of R.Montgomery
 contains
first example (1991) of strictly abnormal sub-Riemannian length-minimizer. Other examples and the corresponding theory of abnormal minimizers can be found in    \cite{AS1,AS2,AS3, Mon2,SuC,SL}.


The problem of finding minimal
admissible path locally in a neighborhood of $q^{0}$
can be represented as
the following Lagrange problem of the Calculus of Variations:
\begin{eqnarray}
& \ell_{1}(u(\cdot))=\int_{0}^{1} \langle u(\tau),
u(\tau) \rangle^{1/2} d\tau
\longrightarrow \min ,                   \nonumber   \\
& \dot{q}=G(q)u, \ q(0)=q^{0}, \ u \in  R^{r},          \label{sys}   \\
& q(T)=q^{1}.                                         \label{ep}
\end{eqnarray}
Here $\langle \cdot , \cdot \rangle$
stays for the inner product in the tangent spaces
$T_{q}M;$ the 'control parameter' $u$ belongs to
$ R^{r};$ the 'controls'  $u(\tau)$ are integrable;
$G(q)=(g^{1}(q), \ldots ,g^{r}(q))$ is a $r-$tuple of smooth
vector fields, which form an orthonormal  basis of
the distribution~$\cal D$.
%%As long as our consideration
%%concerns  a small neighborhood of a
%%non selfintersecting path
%%on $M$,  such a basis can always be  chosen.

It is easy to prove that the
length-minimization problem
is equivalent to the minimization
of  the functional
\begin{equation}                                   \label{min2}
\ell(u(\cdot))=\int_{0}^{1} \langle u(\tau),
u(\tau) \rangle d\tau
\longrightarrow \min .
\end{equation}

Indeed by virtue of Cauchy-Schwartz inequality
$$\ell_{1}(u(\cdot))=\int_{0}^{1} \langle u(\tau),
u(\tau) \rangle^{1/2}d\tau \leq (\int_{0}^{1} \langle u(\tau),
u(\tau) \rangle d\tau)^{1/2}=(\ell(u(\cdot)))^{1/2},$$
and the equality holds if and only if $\|u(\tau)\| \equiv \const$.
As far as the functional of length $\ell_{1}$ does {\em not} depend
on a parameterization of a curve,  we may always assume that
$\|u(\tau)\| \equiv \const$  for a chosen parameterization.
This implies that the minimizers of the functional $\ell$
are the same as the minimizers of $\ell_{1}$. At the same time we
have established that minimizers of the problem
(\ref{sys})-(\ref{min2}) have constant magnitude.




Let us introduce first-order
necessary minimality condition
for the problem (\ref{sys})-(\ref{min2}) - the {\em Pontryagin Maximum Principle}
%%which in this case can be represented as a
%%Hamiltonian form of the geodesic  equation
for the Lagrange problem of
the Calculus of Variations .

\newtheorem{pmp}{Theorem}[subsection]
\begin{pmp}
\label{pont}{\em \small\bf (Pontryagin Maximum Principle)}
If $\hat{u}(\cdot)$ is weak minimizer
for the problem (~\ref{sys})-(\ref{min2}), i.e.
corresponding trajectory $\hq(\tau) \ (\tau \in [0,1])$
of (\ref{sys}), satisfying the boundary conditions
$\hq(0)=q^{0}, \ \hq(T)=q^{1}$,
is $W^{1}_{\infty}-$locally minimal admissible path,
then there exists a nonzero pair
$(\hat{\psi}_{0},\hat{\psi}(\cdot))$,
where $\hp_{0}$ is a nonpositive constant and $\hat{\psi}(\tau)$
is an absolutely
continuous covector-function on $[0,1],$ such that
$\hp(\tau) \in {\cal T}^{*}_{\hq (\tau)}M$
and the quadruple
$(\hu(\cdot),\hq(\cdot),\hp_{0},\hat{\psi}(\cdot))$:  \\
1) satisfies  Hamiltonian system
associated with   a Hamiltonian
\begin{equation}
H(u,q,\psi_{0},\psi)=
\hp_{0} \langle u,u\rangle+
\psi \cdot G(q)u;                              \label{nH}
\end{equation}
in local coordinates on $M$ this system has a form
\begin{equation}
\dot{q}=\partial H/ \partial \psi,    \label{3.4}
\end{equation}
\begin{equation}                     \label{3.41}
\dot{\psi}=-\partial H/ \partial q,
\end{equation}


2) meets stationarity condition  for almost all
$\tau \in [0,1]$:
\begin{equation}
\frac{\partial H}{\partial u}                    \label{3.6}
|_{(\hu(\tau),\hq(\tau),\hp_{0},\hp(\tau))}v
=2\hp_{0}\hu(\tau)+\hp(\tau)G(\hq(\tau))=0,
\end {equation}
and for almost all  $\tau \in [0,1]$
\begin{equation}                                 \label{3.7}
H(\hu(\tau),\hq(\tau),\hp_{0},\hat{\psi}(\tau))=
\hp_{0} \langle \hu(\tau),\hu(\tau) \rangle+
\hp(\tau) \cdot G(\hq(\tau))\hu(\tau)=h_{0}=\mbox{const}.
\ \Box
\end{equation}
\end{pmp}

\newtheorem{geod}{Definition}[subsection]
\begin{geod}
\label{geo}
An extremal of the
Lagrange problem (\ref{sys})-(\ref{min2}), i.e. a
quadruple $\ex$ meeting the conditions of the Theorem
\ref{pont} is called sub-Riemannian geodesic.
Sub-Riemannian geodesic
is called normal, if $\hp_{0} < 0$, and
abnormal, if $\hp_{0}=0.\ $ The corresponding triple $\pex$ is called
sub-Riemannian geodesic path. $\Box$
\end{geod}

{\bf Remark.} It follows immediately from the relations
(\ref{3.6})-(\ref{3.7}) of this  theorem, that
$\hp_{0}\langle \hu(\tau),\hu(\tau) \rangle = \mbox{const}$.


{\bf Remark.} Obviously any  restriction $\rex$  of
normal or abnormal sub-Riemannian geodesic
$\ex$
onto a subinterval
$[0,t] \subset [0,1]$ is also normal or abnor\-mal
sub-Riemannian geodesic correspondingly. $\bbox$

{\bf Remark.} A geodesic path $\pex$ may correspond to
different geodesics with different $\hp_{0},\hp(\cdot). \ \bbox$

In a normal case one may assume $\hp_{0}=-1/2$. Then
(\ref{3.6}) becomes
\begin{equation}
\hu(\tau)=\hp(\tau)G(\hq(\tau)).
\end{equation}



\newtheorem{norcor}[geod]{Definition}
\begin{norcor}
A corank
of a geodesic
path $\pex$
is the dimension of the space of pairs $(\hp_{0},\hp(\cdot)),$ which together
with $(\hu(\cdot),\hq(\cdot),T)$ satisfy Theorem \ref{pont}. $\Box$
\end{norcor}

%%\newtheorem{abcor}[geod]{Definition}
%%\begin{abcor}
%%A geodesic path $\pex$ is called corank $k$ abnormal geodesic path
%%if the space of
%%pairs $(0,\hp(\cdot))$,  which together with $\pex$ satisfy the
%%Theorem~\ref{pont}, is $k-$dimensi\-onal.  $\Box$
%%\end{abcor}

%%{\bf Remark.} One should take precautions,
%%when determining corank of abnormal
%%geodesic path, since in a $k-$dimensional linear space of %%pairs
%%$(\hp_{0},\hp(\cdot))$ there is $k-$ or $(k-1)$-%%dimensional subspace
%%of pairs with vanishing $\psi_{0}$. Therefore it may %%happen, that
%%corank $k$ geodesic path is corank $(k-1)$ abnormal
%%geodesic path. $\bbox$

If a geodesic is abnormal, then the length functional
$\ell$ does not appear in the mini\-mality conditions, provided  by the
Theorem \ref{pont}. No surprise that corresponding geodesic paths
have not too much to do with the sub-Riemannian metric and
the minimality of length. They rather correspond to some extremality of the distribution. It turns out that
the abnormal geodesics
'often'  exhibit a phenomenon called in \cite{Y}
{\em rigidity}.





First  let us define  $W_{1, \infty}$-neighborhood of the given Lipschitzian path %
$t \mapsto  \hq(t), \ (t \in [0,T])$ with fixed terminal points.


   Let us consider the graph %
   $(t,\hq(t)):[0,T] \rightarrow [0,T] \times M$ of this path. %
   In the sufficiently small neighborhood
  $\Omega$  of this graph in $R \times M$ %
   we can choose a %
   basis $B_{t,q}:T_{q}M \rightarrow  R^{n}$ of $T_{q}M$ %
   continuously depending on $(t,q) \in W$.  %
   Then any  Lipschitzian path $q(\cdot)$ on   $M$ %
   parameterized by $[0,T]$ %
   corresponds to a $ R^{n}$-valued vector function %
   $t \mapsto B_{t,q(t)}\dot{q}(t)$ %
   defined almost everywhere on $[0,T].$ %
   We shall identify $\dot{q}(t)$ with $B_{t,q(t)}\dot{q}(t).$ %
   Then we can define %
   a $W_{1,\infty}$-metric or norm %
   in a small $C^{0}$-neighborhood  of $\hq(\cdot)$
in the space of paths with fixed terminal points
by putting: %
   $$\|q^{1}(\cdot)-q^{2}(\cdot)\|_{1,\infty}=\mbox{vraisup}_{t \in [0,T ]} \|\dot{q}^{1}(t)-\dot{q}^{2}(t)\|.$$ %
    %

   \newtheorem{w1i}[geod]{Definition}%
   \begin{w1i}                            \label{def1} %
   An  admissible path %
   $t \mapsto \hq(t), \ t \in [0,T]$ %
   of the distribution $\cal D$ with the end-points %
   $q^{0}$ and $q^{1}$ is $W_{1,\infty}$-local or weak minimizer  if, %
   for some neighborhood of $\hq(\cdot)|_{[0,T]}$ %
   in $W_{1,\infty}[0,T],$ %
   the points $q^{0}$ and $q^{1}$ can not be connected by %
   a shorter admissible path %
$t \mapsto \bar{q}(t), \  t \in [0,T]$ %
   belonging to this neighborhood. $\Box$ %
   \end{w1i} %



\newtheorem{rigi}[geod]{Definition}
\begin{rigi}
\label{rig}
An admissible path $q(\cdot)$ of the vector distribution $\cal D$
with end-points  $q^{0}$ and $q^{1}$ is called rigid if
it is isolated up to a reparameterization
in the metric of $W_{1,\infty}$ in the set
${\cal P}_{q^{0}}^{q^{1}}$
of all admissible paths, which connect $q^{0}$ and $q^{1}.\ \Box$
\end{rigi}


Rigid admissible paths are formally weakly minimal.
 Theorem \ref{pont}  provides
necessary conditions of weak minimality. When it is applied to rigid paths then  in addition
one should  take  $\hp_{0}=0$. This implies that
if an admissible path $(\hu(\cdot),\hq(\cdot))$ is rigid on $[0,T]$, then $\pex$ is an abnormal geodesic path.


We dealt with  the rigidity  in \cite{AS1,AS2};
the readers may consult  that paper for   detailed study of
the phenomenon.






%%%{\bf Remark.} As it is known (\cite{HH}), admissible
%%paths (without or with pre given end-points)
%%of a completely nonholonomic vector distribution $\cal D$ are dense
%%in metric of $C^{0}$ in the space of all paths on $M$
%%(correspondingly without or with pre given end-points). Therefore an
%%admissible path is never isolated in the metric of $C^{0}$.
%%Therefore  strong (=$C^{0}-$local) minimality for
%%sub-Riemannian geodesics need different treatment.
%%We are going to perform results on strong minimality
%%of abnormal sub-Riemannian geodesics
%%for 2-dimensional vector distributions
%%in a forthcoming paper.$\bbox$


To finish with the first-order condition
given by the Theorem \ref{pont} let us note that in the
abnormal case the Hamiltonian (\ref{nH}) degenerates into
an abnormal Hamiltonian
\begin{equation}                           \label{abH}
H=\psi \cdot G(q)u.
\end{equation}
The stationarity condition (\ref{3.6})
for an abnormal geodesic
amounts to the orthogonality of $\hp(\tau)$ to the
distribution $\cal D$ at every
point $\hq(\tau):$
\begin{equation}                                      \label{ortd}
\hp(\tau) \cdot G(\hq(\tau))=0,  \
\forall \tau \in [0,1].
\end{equation}

\subsection{Second Variation along Normal and Abnormal Geodesics}
\label{svar}

In the previous Section we have reduced
the problem of finding minimal admissible
%%%%(=tangent to the distribution
%%%%$\cal D$)
path between given points $q^{0}$ and $q^{1},$
to the Lagrange problem (\ref{sys})-(\ref{min2}).
We have formulated first-order necessary condition of weak minimality
 and defined  sub-Riemannian geodesics.  The solutions of the length minimization problem
should be sought among geodesic paths.
In this Section we are going to introduce
first and second variation along  a geodesic $\ex$.




%%Let us start with definitions of first and second variations along an
%%abnormal geodesic $\abex$.
%%Everywhere in this Section we assume, that $\hu(\cdot)$
%%is continuous (from the left) at $T.$
%%We choose $\bar{T}>T$ and put $\hu(t)=\hu(T)$ on $[T,\bar{T}].$
Let us introduce
an {\em input/state map}
$\Phi_{t}:R \times L^{r}_{\infty}[0,1] \ar M,$
which maps an  input   $u(\cdot)$ into the point
$q(t)$  of the trajectory $q(\cdot)$ of the
system $\dot{q}=G(q)u(\tau), \ q(0)=q^{0}.$
Obviously, $\Phi_{t}(\hu(\cdot))=\hq(\cdot)$ and
$\Phi_{1}(\hu(\cdot))=\hq(1)=q^{1}.$

Let $\hat{P}_{t}$ be  the  flow generated by the reference vector field
$\hf_{t}(q)=G(q)\hu(t)$.
Let us define a 'pull-backed' input/state  map
$F_{t}(u(\cdot))=\hat{P}_{t}^{-1}(\Phi_{t}(u(\cdot))$ and
put $F(u(\cdot))=F_{1}(u(\cdot))$. Obviously $F(\hu(\cdot))=q^{0}$.

We put
$$\ell (u(\cdot))=\int_{0}^{1} \langle u(\tau),u(\tau) \rangle d\tau; \
\ell : L^{r}_{\infty}[0,1] \rightarrow  R.$$


A well known fact is that for
$\hu(\cdot) \in  L^{r}_{\infty}$ to
be a weak minimizer for the Lagrange problem
(\ref{sys})-(\ref{min2}) it
must be a critical point of the map
$(\ell,F).$
Indeed otherwise  the Implicit Function Theorem
implies existence  (locally,  in a $L_{\infty}$-neighborhood of $\hu(\cdot)$) of solutions of
 the system of equations
\[ \ell(u(\cdot))=\ell(\hu(\cdot))-\epsilon,
  \  F(u(\cdot))=q^{1}, \]
 for any sufficiently small
$\epsilon >0$.
This means that  $q^{0}$ and $q^{1}$ can be connected by
an admissible path of length
$\ell(\hu(\cdot))-\epsilon<\ell(\hu(\cdot)).$

If  $\hu(\cdot)$ is a critical point
for the map  $(\ell,F),$
i.e. the differential
$(\ell ',F')|_{\hu(\cdot)}:L^{r}_{\infty}
\rightarrow  R \times {\cal T}_{q^{0}}M$
is not onto, then there exists a pair
$(\hp_{0},\hp_{T}) \in  R \times {\cal T}^{*}_{q^{0}}M,$
which annihilates the image of
$(\ell ',F')|_{(\hu(\cdot))}$:
\begin{equation}                    \label{crit}
\hp_{0}\ell '+\hp_{T}F' \equiv 0.
\end{equation}
This equality is equivalent to the statement of the Theorem \ref{pont}
with $\hp_{T}$ being the initial  value $\hp(0)$ for the solution of
the adjoint equation (\ref{3.41}).
If $\hp_{0}~=~0,$ then the functional $\ell$ appears neither in (\ref{crit}) nor in the Theorem \ref{pont}.
In this case
$\hu(\cdot)$ is a cri\-ti\-cal point of the map  $F$, or equivalently  is an abnormal geodesic control  corresponding  to the geodesic $(\hu(\cdot),\hq(\cdot),0,\hp(\cdot))$.  0

To study the geodesics -
the critical points of $(\ell ,F)$ -
we have to invoke
initial  terms of  Taylor expansion for $F.$
Let us present $F(u(\cdot))$ as a composition of
chronological exponentials  (see Section 2 for the notation):
$$\chro_{0}^{1}Gu(\tau)d\tau \circ
\left(\chro_{0}^{1}G\hu(\tau)d\tau\right)^{-1},$$
acting on the  point $q^{0}$.
Putting $u(\tau)=\hu(\tau)+v(\tau)$ and applying  the
variational formula (\ref{P4}) to the first
of the chronological exponentials we obtain
$$\chro_{0}^{1}G\left(\hu(\tau)+v(\tau)\right)d\tau =
\chro_{0}^{1}Y_{\tau}v(\tau)d\tau \circ \chro_{0}^{1}G\hu(\tau)d\tau ,$$
where
$$Y_{\tau}v=\Ad  \chro_{0}^{\tau} G \hu(\xi)d\xi Gv.$$
Therefore
\begin{equation}                \label{3.18}
F(u(\cdot))=\chro_{0}^{1}Y_{\tau}v(\tau)d\tau (q^{0}).
\end{equation}
%%%From the formula (\ref{P66}) it follows that
%%%\begin{equation}                              \label{add}
%%%dY_{\tau}/d \tau=-\ad \hf_{t} Y_{t,\tau}.
%%%\end{equation}

%%%Putting
%%%\begin{equation}                                  \label{Y1}
%%%Y_{\tau}v=Y_{T,\tau}v=
%%\Ad \chro_{T}^{\tau} G \hu(\xi)d\xi Gv,
%%%\end{equation}
%%% we compute (compare with \cite{AS1})
Invoking the Volterra expansion of the chronological exponential (\ref{3.18}) we obtain the following expression for the first differential of $F$ at $\hu(\cdot)$:
\begin{equation}
F'|_{\hu(\cdot)}(u(\cdot))=             \label{4.0}
 \int_{0}^{1} Y_{\tau}(q^{0}) u(\tau)d\tau .
\end{equation}


If $\hu(\cdot)$ is a critical point of $(\ell,F)$,
then
$\im (\ell',F')|_{\hu(\cdot)} \neq  R \times {\cal T}_{q^{0}}M, $
and there exists a nonzero covector
$(\hp_{0},\hp_{T})  \in {\cal T}_{q^{0}}^{*}M,$
which annihilates
$\im (\ell',F')|_{\hu(\cdot)}$:
$$\int_{0}^{1} \left(2\hp_{0}\hu(\tau)+
\hp_{T} \cdot Y_{\tau}(q^{0})\right)u(\tau)d\tau =0,$$
for all $u(\cdot) \in L^{r}_{\infty}[0,1]$.
The latter  equality implies:
\begin{equation}
2\hp_{0}\hu(\tau)+\hp_{T} \cdot                        \label{3.11}
Y_{\tau}(q^{0})=0, \
\mbox{for almost all}\  \tau \in [0,T].
\end{equation}

This  condition
is  equivalent to the condition (\ref{3.6})
of the Theorem~\ref{pont}.
Namely if one chooses  the solution $\hp(\cdot)$
of the adjoint equation
(\ref{3.41}) with the initial  value $\hp(0)=\hp_{T},$
then  $\hp(\cdot)$  satisfies the stationarity
condition (\ref{3.6}).
One defines {\em corank of the geodesic path}  $\pex$
as   the corank
of $(\ell',F')|_{\hu(\cdot)}$.

\newtheorem{firvar}{Definition}[subsection]
\begin{firvar}
The first differential
$(\ell',F')|_{\hu(\cdot)}:L^{r}_{\infty} \rightarrow
{\cal T}_{q^{0}}M,$
at a critical 'point' $\hu(\cdot)$
is called first variation
along the geodesic path $\pex .$
It is computed according to the
formula (\ref{4.0}).
$\Box $
\end{firvar}

In  what follows we assume   either $\hp_{0}=-1/2$ or $\hp_{0}=0$.

Now  we introduce the {\em second variation}
along a  geodesic $\abex$.
It is the  Hessian, or quadratic differential
of $(\ell,F)$, at the critical point
$\hu(\cdot) \in  L^{r}_{\infty}$
(see \cite{AVG}).
Choosing a function $\chi :M \longrightarrow  R,$ such that
$d\chi |_{q^{0}}=\hp_{T},$
let us consider a function
$\phi(u(\cdot))=\hp_{0}\ell \left(u(\cdot))+\chi(F(u(\cdot))\right).$
Since the pair $(\hp_{0},\hp_{T})$
annihilates $\im (\ell',F')|_{\hu(\cdot)},$
then $\hu(\cdot)$ is a critical point for this function.

Let us compute the quadratic term of Taylor expansion
for $\phi(t,u(\cdot))$ at  $(T,\hu(\cdot)).$
Appealing to the Volterra expansion
(~\ref{P2}) for the right chronological exponential (\ref{3.18}), we
obtain
\begin{equation}                     \label{4.1}
\phi ''|_{\hu(\cdot)}(u(\cdot))=
\hp_{0}\int_{0}^{1}\langle u(\tau), u(\tau) \rangle d \tau +
\left(\int_{0}^{1}Y_{\tau}u(\tau) \circ \int_{0}^{\tau}Y_{\xi}u(\xi)d\xi)d \tau  \chi)
(q^{0}\right) .
\end{equation}
%%%(When carrying the computation one should take into account
%%%the equalities (\ref{3.7}), (\ref{add}) and (\ref{3.11})).

When restricting the quadratic
form (\ref{4.1}) to the kernel of $(\ell',F')|_{\hu(\cdot)},$
determined by the equalities
\begin{equation}                           \label{dom2}
\int_{0}^{1}\hu(\tau) u(\tau) d \tau =0, \
   \int_{0}^{1}   Y_{\tau}(q^{0}) u(\tau) d \tau =0,
\end{equation}
we are
able to subtract from (\ref{4.1}) a vanishing value of
\[ \frac{1}{2}
(\int_{0}^{1}Y_{\tau}u(\tau)d\tau)
\circ \int_{0}^{1}Y_{\tau}u(\tau)d\tau)
\chi)(q^{0}) ,  \]
and transform (\ref{4.1}) into
\[\hp_{0}\int_{0}^{1}\langle u(\tau), u(\tau) \rangle d \tau +
 \frac{1}{2}\left((\int_{0}^{1}
[ \int_{0}^{\tau}Y_{\xi}u(\xi)d\xi,
Y_{\tau}u(\tau)]d\tau)
\chi \right)(q^{0}). \]


The last expression does not depend on a choice of $\chi$ but
only on $\hp_{T}=d\chi |_{q^{0}}$ and therefore we may introduce the following definition.

\newtheorem{modif}[firvar]{Definition}
\begin{modif}
\label{sevar}
The second variation along the geodesic
$\abex$ is the quadratic form
\begin{equation}                                           \label{4.2}
Q[\hp_{0},\hp_{T}]
(u(\cdot))=\hp_{0}\int_{0}^{1}\langle u(\tau), u(\tau) \rangle d \tau
+ \hp_{T}\cdot
\int_{0}^{1}
\left[\int_{0}^{\tau}Y_{\xi}u(\xi)d\xi,
Y_{\tau}u(\tau)\right](q^{0})d\tau ,
\end{equation}
whose domain is subspace of $L^{r}_{\infty}$
defined by the conditions (\ref{dom2}), and either
$\hp_{0}=-1/2$ (normal case)
or $\hp_{0}=0$ (abnormal case).  $\Box$
\end{modif}


\section{Morse Index
and Necessary Conditions for Weak Mi\-ni\-ma\-lity.}
\label{morse}
\setcounter{equation}{0}

In this Section we define an important invariant of a geodesic - {\em Morse index}. Then we establish its interrelation with weak minimality of geodesics and provide
necessary condition for finiteness of index for abnormal geodesics.


\newtheorem{inde}{Definition}[section]
\begin{inde}
\label{inde}
Morse index of the geodesic $\ex$ is positive index of the
quadratic form (\ref{4.2})-(\ref{dom2}),
i.e. maximal among the dimensions of the subspaces
in its domain, on which the quadratic form is positive definite.
$\ \Box$
\end{inde}

\newtheorem{indpa}[inde]{Definition}
\begin{indpa} \label{indp}
Morse index of  the geodesic path $\pex$
is minimum of indices of the
geodesics $\ex$,
which correspond to this geodesic path, or minimum
of the positive indices of quadratic forms $Q|_{\hu(\cdot)}[\hp_{0},\hp_{T}]$
for all possible
$(\hp_{0},\hp_{T}) \perp \im F'|_{\hu(\cdot)}. \ \Box$
\end{indpa}

We now set second-order necessary weak minimality condition for
corank $k$   geodesics paths.
It is  a corollary of a sufficient condition
for stable  local  openness  of a smooth map in its
 critical point (Theorem~\ref{wmn}).



%%Corresponding result for corank $1$
%%case was established in \cite{AS1}.

%%%\newtheorem{finind}{Theorem}[section]
%%\begin{finind}[Necessary Rigidity Condition for Abnormal Geodesics]
%%\label{abne}
%%For a corank $k$ abnormal geodesic path
%%$\pex$
%%to be rigid its index
%%should not exceed $k-1$.
%%In particular index of a rigid corank
%%$1$ abnormal geodesic path must vanish. $\Box$
%%\end{finind}

%%Generally rigidity is stronger than weak minimality. But %%whenever
%%all geodesics, which a geodesic path $\pex$ enters, are %%abnormal,
%%then the conditions of the Theorem \ref{abne}
%%are necessary for weak minimality of the path.
%%It follows from the Propositions \ref{rep} and \ref{quad}
%%(see Appendix).


\newtheorem{nemin}{Theorem}[section]
\begin{nemin}[Morse Index and
Weak Minimality of Geodesic Paths]
\label{nemi}
Morse index of a weakly minimal corank $m$ geodesic path $\pex$
must be $\leq m-1. \ \Box$
\end{nemin}


This theorem  can be  derived  from a general result  on stable local openness  of smooth maps   of Banach spaces at their critical points
(stable local openness of a map at  its regular point
is  corollary of the Implicit Function Theorem).


Let a map $G:X \rightarrow  R^{k}$ be
twice Frechet differentiable at a  point $\hx$
which is critical point of $G$.
Let us introduce the Hessian of $G$ at the point $\hx$ ,
which is the quadratic map
$G''(\hx):\ker G'|_{\hx} \rightarrow \coker G'|_{\hx}$
(see \cite{AVG}).
One can represent it as a bundle of quadratic forms
$$ \lambda \mapsto  \lambda G''(\hx)(\xi , \xi), \
\xi \in \ker G'|_{\hx}, \ \lambda \perp  \im  G'|_{\hx}.$$
We define {\em index} and {\em nullity} of
$(\hx ,\lambda)$ as positive index and dimension of the kernel
of the  quadratic form $\lambda G''(\hx)(\xi , \xi)$.
We define {\em index of critical point} $\hx$ as
$$\min\{\ind \lambda G''|\lambda \perp  \im  G'|_{\hx}\}.$$



\newtheorem{wmne}[nemin]{Theorem}
\begin{wmne}%
{\em {\small\bf
(Morse Index and Stable  Local Openness of a Smooth Map)}}
\label{wmn}
Let $X$ be  a Banach space and
$\hx \in X$ be a corank $m$ critical point
for the map  $G:X \rightarrow  R^{k}$,
which is twice Frechet differentiable at $\hx$.
If the Morse index of $\hx$ is $\geq m$, then the map $G$
is  locally  open   in a stable way
i.e. for any neighborhood
$U$ of $\hx \in X$ there exists a $C^{0}$-neighborhood
${\cal V}$ of $G$ such that for any $\bar{G} \in {\cal V}: \
G(\hx) \in \vnu \bar{G}(U). \ \Box$
\end{wmne}

 Some version  of this result has been proved   in \cite{A4}.
As far as this source is not very accessible
we provide in Subsection~\ref{losur} a sketch of  the   proof.

Now  basing on this result we provide

{\em Proof of the Theorem}~\ref{nemi}. Let us consider the Banach space
$X=L_{\infty}^{r}[0,1]$ and the
map  $G=(\ell,F):L_{\infty} \ar  R \times M$, where $\ell ,F$
are as defined above. If $\hu(\cdot)$ is a geodesic  control,
i.e. a critical point
of $G$ of corank $m$,  and Morse index of this
geodesic control is $\geq m$, then,
according to the Theorem~\ref{wmn},  $G\left(\hu(\cdot)\right) \in \vnu G(U)$
for any neighborhood $U$ of $\hu(\cdot)$ in $L_{\infty}$. Therefore
there  exists $\bar{u}(\cdot) \in U$ such that
$\ell\left(\bar{u}(\cdot)\right) <  \ell \left(\hu(\cdot)\right), \
F\left(\bar{u}(\cdot)\right) =  F \left(\hu(\cdot)\right)=q^{1}$. i.e. the path
generated by the control $\bar{u}(\cdot)$ connects $q^{0}$ with
$q^{1}$ and is {\em strictly shorter}
then the reference path. $\Box$





From the Theorem \ref{nemi} it follows, that
{\em finiteness of Morse index
is necessary for weak mi\-ni\-ma\-lity}
of a geodesic.
There is a classical result which claims that
{\em for normal geodesics Morse index is finite}.
We are going to introduce
a necessary condition for the  finiteness of index
for abnormal geodesics.


%%Denoting again by $V_{\hu(\tau)}^{\perp}$
%%the orthogonal complement to $\hu(\tau)$ in $ R^{r}$ we
%%introduce first of these conditions:
%%for almost all $\tau \in [0,T]$
%%\begin{equation}                      \label{goh}
%%\hp(\tau) \cdot [Gv,Gw](\hq(\tau)) =0
%%\ \forall v,w \in \Vp.
%%%\end{equation}

\newtheorem{goco}[inde]{Definition}
\begin{goco}[Goh necessary condition]
\label{goc}
An abnormal geodesic satisfies Goh condition if
for almost all $\tau \in [0,1]$
\begin{equation}                                  \label{stgoh}
\hp(\tau) \cdot [Gv,Gw](\hq(\tau)) =0,
\ \forall v,w \in  R^{r},
 \end{equation}
i.e. $\hp(\tau)$ annihilates the Lie square
$\cD^{2}(\hq(\tau))$ of the distribution $\cal D$
at almost every point of the geodesic path $\hq(\cdot). \ \Box$
\end{goco}


For singular extremals of optimal control problems this condition has been
introduced by B.S. Goh in \cite{Goh}.
One can find in \cite{AS2} the proof
of the following result for abnormal geodesics  corresponding to
arbitrary measurable bounded geodesic control $\hu(\cdot)$.

\newtheorem{gone}[nemin]{Proposition}
\begin{gone}%
{\em {\small\bf
(Goh Condition and Finiteness of Morse Index
for  Abnormal Geodesics)}}
\label{gon}
For index of an abnormal geodesic to be finite the Goh condition
must hold  along it. $\Box$
\end{gone}




Now we are able to formulate a condition of
weak minimality  for
{\em strictly} abnormal geodesic paths, i.e. such that
all the
corresponding geodesics are abnormal.

\newtheorem{legomin}[nemin]{Theorem}
\begin{legomin}
\label{legom}
{\em \small\bf (Necessity of Goh  Condition
for  Abnormal Weak Minimality)}
Let $\pex$ be   strictly  abnormal geodesic path.
If  this path is
weak length   minimizer then the Goh condition (\ref{stgoh})
must hold for  some of the corresponding (abnormal)
geodesics, i.e.  for some choice of $\hp(\cdot)$.
$\Box$
\end{legomin}

{\em Proof of the Theorem}~\ref{legom}. It has been established  in
\cite[Appendix 1]{AS2}  (see also proof of the Lemma~\ref{lwe} below)
 that if  Goh condition is not satisfied
along an abnormal geodesic than Morse index of this geodesic is infinite.
If all geodesics corresponding to the weakly minimal
geodesic path $\pex$ are abnormal, then according to the
Theorem~\ref{nemi} Morse index of at least one of them must be finite and
therefore the Goh condition (\ref{stgoh}) must hold along this abnormal geodesic.
$\Box$

An immediate corollary of this theorem is that
\begin{center}
{\em if a germ of distribution
is $2$-generating, i.e. $\cD^{2}(q^{0})=T_{q^{0}}M$, then there are no weakly
minimal strictly abnormal geodesic paths  starting at}  $q^{0}$.
\end{center}

It turns out that the same is true for
a bigger  class   of {\em medium fat}  distributions.

\newtheorem{mefami}[nemin]{Theorem}
\begin{mefami}
\label{mefa}{\em \small\bf (Lack  of Abnormal Weak Minimizers
for Medium Fat Distributions)}
If a germ of distribution
is medium fat, i.e.
$$\forall X \in \cD, \ X(q^{0}) \neq 0: \
[X,\cD^{2}](q^{0})+\cD^{2}(q^{0})=T_{q^{0}}M,$$
then there are no weakly
minimal strictly abnormal geodesic paths  starting at $q^{0}. \ \Box$
\end{mefami}
The proof of this result is provided in the subsection~\ref{noab}.





%%\end{document}
\section{Subanalyticity of Sub-Riemannian Metrics for
Medium Fat and 2-Generating Distributions}
\label{ssrs}
\setcounter{equation}{0}

It turns out that the same conditions
which rule out
 existence of abnormal weak length minimizers
for some classes of distributions
guarantee subanalyticity of  germs of  sub-Riemannian
 metrics for these classes of distributions.


\newtheorem{mefasu}{Theorem}[section]
\begin{mefasu}
\label{mefas}{\em \small\bf (Subanalyticity of sub-Riemannian Metrics for Medium Fat Distributions)}
Let $\cD$ be a germ at  $q^{0} \in M$  of distribution which is
medium fat at this point. Then    small sub-Riemannian
$\rho$-spheres
$\{q \in M | \ \rho_{\cD}(q^{0},q) =\rho >0 \}$
are subanalytic subsets of $M$ for all sufficiently small
$\rho >0$.
Moreover the sub-Riemannian metric
$\rho_{\cal D}(q,q')$ is subanalytic in a small neighborhood of $(q^{0},q^{0})$ in $M \times M$ outside the dia\-go\-nal $\{(q,q)\}.   \ \Box$
\end{mefasu}


One can say more if the distribution
$\cal D$ is 2-generating.

\newtheorem{2Dsub}[mefasu]{Theorem}
\begin{2Dsub}
\label{2Dsu}{\em \small\bf (Subanalyticity of sub-Riemannian Metrics for 2-generating Distributions)}
Let $\cD$ be a germ at $q^{0} \in M$ of a distribution which is
2-generating  at this point. Then the sub-Riemannian metric
$\rho_{\cal D}(q,q')$ is subanalytic in a small neighborhood of $(q^{0},q^{0})$ in $M \times M. \ \Box$
\end{2Dsub}

The crucial fact for  establishing  the subanalyticity  is the
one  that the length-minimizing geodesics $\ex$ can be parameterized by a {\em compact} set.

Let us recall that according to the Pontryagin Maximum Principle (Theorem~\ref{pont}) any  normal (with $\hp_{0}=-1/2$)
geodesic control $\hu(\cdot)$
is computed as $\hu(t)=\hp(t)G(\hq(t))$.
Assume that all geodesics
are parameterized by the interval $[0,1]$.
Then the constant value $h$ of the Hamiltonian (\ref{nH})
along this geodesic
equals  $\|\hp(t)G(\hq(t))\|^{2}/2$.
Hence $\hu(\cdot)$ has constant magnitude
 $\|\hu(t)\| \equiv \sqrt{2h}$ on $[0,1]$ and the
length of the geodesic path equals  $\rho=\sqrt{2h}$ as well.


Obviously each geodesic is determined by the initial value
$\hp_{T}=\hp(0)$ of $\hp(\cdot)$.
As  far  as for a geodesic of fixed length $\rho$ one has
$\|\hp(t)G(\hq(t))\| \equiv  \rho$,
then we may think  that  all initial values $\hp_{T}$ have
 horizontal
component of norm $\rho$.
Thus the space of $\hp_{T}$
can be identified with the  cylinder $S^{r-1} \times  R^{n-r}$.
Our goal is proving  that
\begin{center}
the length-minimizing geodesics  are
parameterized by $\hp_{T}$ from a compact part $K$  of the cylinder.
\end{center}

We shall also need the asymptotic
dependence of this compact part on the value of $\rho$, as
$\rho \rightarrow +0$.


\newtheorem{compac}[mefasu]{Theorem}
\begin{compac}
\label{compa}{\em \small\bf (Compactness of the Set of Minimal Geodesics
for 2-Generating and Medium Fat Distributions)}
Assume that the geodesics are  parameterized by the interval
$[0,1]$. Then for some $\rho_{0}>0$:
\begin{itemize}
\item[i)]  If $\cD$ is a germ at $q^{0}$ of  medium fat
distribution then for  any $q$ from the sub-Riemannian
$\rho_{0}$-neighborhood of $q^{0}$
the set
of minimal geodesics, starting at $q$,
of   given length $\rho < \rho_{0}$
is compact (parameterized by a compact set of $\hp_{T}$, the same for all $q$);
\item[ii)]   if this germ
is  $2$-generating then there exists
$A>0$ such that for all the
 length-minimizers of  length $\rho < \rho_{0}$
   there holds $\|\hp_{T}\| \leq A$;

\item[iii)]  if the germ
is   medium fat then for any $\alpha >3$  there exist
$\bar{A}>0$ such that for all
 the  length-minimizers
of positive length
$\rho \leq \rho_0$ there holds $\|\hp_{T}\| \leq \bar{A}\rho^{-\alpha -2}. \ \Box$
\end{itemize}
\end{compac}





The subanalyticity claimed in the theorems~\ref{mefas} and \ref{2Dsu}  follows from this latter theorem  by a standard reasoning.
We shall prove subanalyticity
of the square of the sub-Riemannian
length $\rho^{2}_{\cal D}$,
which immediately  implies
the subanalyticity
of  the sub-Riemannian
length $\rho_{\cal D}$.

Let us consider the sub-Riemannian exponential map  $E(q, \hp_{T})$ which maps the pair $(q, \hp_{T})$ into the final point $\hq(1)$ of the corresponding geodesic path;
this map is obviously analytic.
For given geodesic path
defined by  $(q, \hp_{T})$
the square of  its length
 equals to
$2h(q, \hp_{T})$ - the doubled value of the (analytic) Hamiltonian
along this geodesic. Then we may calculate $\rho^{2}_{\cal D}(q,q')$ as
$$\rho^{2}_{\cal D}(q,q')=
\min \{2h(q, \hp_{T}): \   E(q, \hp_{T})=q'\}.$$


According to the Theorem~\ref{compa} one can assume
in the $2$-generating case  that $\hp_{T} \in T^{*}_{q^{0}}M$
satisfies $\|\hp_{T}\| \leq A$
provided that
$\rho^{2}_{\cal D}(q,q') \leq \rho_{0}^{2}$.
In the medium fat case
if
$0 < \delta^{2} \leq \rho^{2}_{\cal D}(q,q') \leq \rho_{0}^{2}$,
then $\|\hp_{T}\| \leq \bar{A}\delta^{-\alpha -2}$ for some
$\alpha >3$. In both cases $\hp_{T} \in K, \ K$ - compact.





Consider the set of triples $(q,q',\hp_{T})$
from  $M \times M \times K$ determined
by the relation $E(q, \hp_{T})=q'$. This set is obviously subanalytic,
and the image of this set under the analytic map
$(q,q',\hp_{T}) \mapsto \left(q,q',\hp_{T},2h(q,\hp_{T})\right)$ is subanalytic as well.
The image of the latter set under the projection
$(q,q',\hp_{T},c) \mapsto (q,q',c)$ is subanalytic.
Here is where the compactness of $K$ plays crucial role.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{Indeed  let  $S$
be  a subanalytic subset of
$X \times Y$ ($X,Y \ - \ C^{\omega}$-manifolds). Let $\pi: X \times Y \rightarrow X$ be the projection onto $X$.
If for any compact $C \subset X$ there exists compact $C' \subset Y$ such that
$\forall x \in C: \ \pi^{-1}(x) \subseteq C'$,
then $\pi(S)$ is subanalytic.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This image is    the  epigraph of the
function $\rho^{2}_{\cal D}(q,q')$. Therefore  both this  function and the sub-Riemannian metric are subanalytic.
$\Box$



\section{Generic 2-Generating and Medium Fat Distributions}
\label{gener}
\setcounter{equation}{0}

In the previous Section we  established subanalyticity of
sub-Riemannian metrics for  the classes of 2-generating and medium fat distributions. These classes are substantially larger than the class of
fat distributions.
Here we discuss in what dimensions the properties of   medium fatness or of being 2-generated are generic.


Let us start with the class of fat distributions for which
subanalyticity of sub-Riemannian metrics
has been established  earlier
by H.J.Sussmann (\cite{Sus}) and Zhong Ge (\cite{ZG}).
This class  is
relatively poor; if we  restrict ourselves to
generic distributions then it  basically
involves  germs of corank $1$ distributions on
odd-dimensional  manifolds.

Indeed if we choose a basis
$X^{1}, \ldots ,X^{2k}$ of the $2k$-distribution $\cD$
on a $(2k+1)$-dimensional manifold, and a (unique up to multiplication by a scalar)  covector
$\psi \in T^{*}_{q}M$ which annihilates $\cD_{q}$,
then
for generic $\cD$ the {\em even-dimensional skew-symmetric matrix}
$C=(c^{\psi}_{ij})=\left\langle \psi , [X^{j},X^{i}](q) \right\rangle , \ i,j=1, \ldots , 2k$
is nonsingular.  Hence for  $X=\sum_{i=1}^{2k}X^{i}v_{i}$   the condition
$$[X,\cD](q) \subset \cD(q)
\leftrightarrow
\left\langle \psi , [X,\cD](q) \right\rangle=0,$$
is equivalent to the equality $Cv=0$ where $C$ is defined above
and $v=(v_{1}, \ldots , v_{2k})$. This equality  is
only satisfied when $v=0$.

If one considers a corank $1$
distribution $\cD$ on odd-dimensional manifold,
then repeating the same argument one comes
to an odd-dimensional skew-symmetric matrix $C$, which is   always
singular and therefore always has a nontrivial vector
$v=(v_{1}, \ldots ,v_{n})$ in its kernel. Then
$\left\langle \psi,[\sum_{i=1}^{n}X^{i}v_{i},\cD](q)\right\rangle=0$ and hence the
distribution $\cD$ is {\em not} fat.

It is easy to see that in  a class  of distributions of   given  corank  $>1$
there exists  open (in $C^1$-metric) subset
consisting of nonfat distributions. Indeed  let us take  a pair of linearly-independent
covectors $\psi^{1},\psi^{2}$  from the annihilator
$\cD^\perp$ of a distribution
and construct the   matrices $C^{\psi^{1}},C^{\psi^{2}}$
as above.  If there is a
stable intersection at the 'point'
$C_0=\alpha_{0} C^{\psi^{1}} +(1-\alpha_{0}) C^{\psi^{2}} $ of the pencil
$\alpha C^{\psi^{1}} +(1-\alpha) C^{\psi^{2}} \ (\alpha \in R)$ with the
cone $\{C:| \det C=0\}$, then  taking
any vector $v$ from the kernel of $C_0$,   we construct,
as above, the vector field $X=\sum_{i=1}^{2k}X^{i}v_{i}$
such that $\left(\cD + [X,\cD]\right)(q)$ is annihilated by the covector
$\alpha_0 \psi^{1} + (1 -\alpha_0)\psi^{2}$.

Still in some  classes of germs of corank $c>1$  distributions  there are
open subsets consisting of fat distributions.  The corresponding pairs $(c,n)$ (with $n$ being a  dimension of  manifold) have been characterized by
C.B.Rayner (\cite{Ray}; see also \cite{Mon2}).






The class of $2$-generating distributions is essentially bigger.

 It is known (see \cite{VG}, and  Subsection~\ref{distr})
that for a generic $r$-distribution on $n$-dimensional manifold the initial segment o the  vector of growth
$n_{1} \leq n_{2} \leq \cdots$ is:
$$n_{1}=r,n_{2}=\min \{n,r(r+1)/2\}, \ldots .$$
We are going to prove this result for the manifolds  of dimension
$n \leq r(r+1)/2$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us
take any germ $\cD$ of distribution at $q \in M$
and introduce an adopted coordinate system
$(x,y):{\cal O} \rightarrow  R^n$ such that :

i) $(x,y)(q)=0_{R^n}$,

ii) $(x,y)_*|_{q}f_i(q) \
(i=1, \ldots , r)$ coincide with the first $r$ elements
$e_i , i=1, \ldots ,r,$ of the standard basis of $ R^n$;

iii) the subspace   $\Delta^2_{q}= \span\{\cD_{q},
[f^i,f^j](q), | i < j \}$, is mapped isomorphically by the differential        $x_*|_{q}$  onto a subspace of $ R^n$, while $y_*|_{q}\Delta^2_{q}=0$.

Denote by $\bar{\Gamma}$ a minimal subset of $\Gamma=\{(ij)|i<j\}$
such that
$\Delta^2_{q}$ is spanned by $\cD_{q}$ and
$\{[f^i,f^j](q), | \ (ij) \in \bar{\Gamma} \}$.
 Let
$y=(y_1, \ldots , y_s); \ s =\codim \Delta^2_{q}$.
 Obviously
$$s=\corank \cD_{q}- \sharp \bar{\Gamma}
\leq r(r-1)/2- \sharp \bar{\Gamma}=\sharp(\Gamma \setminus \bar{\Gamma}).$$
Let us choose an arbitrary injection
$\sigma : \{1, \ldots , s\} \rightarrow  \Gamma \setminus \bar{\Gamma}$.   Denote by $\pi$ the projection
$(ij) \mapsto i$ defined on $\Gamma$   and introduce vector   fields
$$\tilde{f}_j=f_j+\sum_{\alpha=1}^s \varepsilon_\alpha x_{\pi(\sigma(\alpha))}
\partial /\partial y_\alpha, \   (\varepsilon_\alpha \in  R , \  \alpha=1, \ldots , s).$$
 For these vector field we establish by direct computation that
$$[\tilde{f}_i,\tilde{f}_{j}](q)=
\left\{
\begin{array}{l}
 [f_i,f_{j}](q), \ \mbox{if
$(ij) \not\in \sigma(\{1, \ldots , s\})$, in particular
if  $(ij) \in \bar{\Gamma}$},  \\
  \lbrack f_i,f_j](q)+
\varepsilon_{\sigma^{-1}(ij)} (\partial/ \partial y_{\sigma^{-1}(ij)})|_q  \ \mbox{otherwise.}
\end{array}
\right.$$
Thus  we constructed a distribution
$\tilde{\cD}$ generated by
$\tilde{f}^1, \ldots , \tilde{f}^r$  which  is
2-generating
and is arbitrarily close to $\cD$ in $C^\infty$ topology if
$|\varepsilon_\alpha|,  \ \alpha=1, \ldots ,s$
are sufficiently small.     $\Box$





Therefore  we may formulate the following conclusion.


\newtheorem{ge2ge}{Proposition}[section]
\begin{ge2ge}{\em \small\bf (Generic 2-Generating Distributions)}
\label{getu}
Generic germ of rank $r$  distribution on $n$-dimensional manifold
with $r \leq n  \leq \frac{r(r+1)}{2}$ is $2$-generating. $\Box$
\end{ge2ge}



%%Thus  for example
%%generic germs of $3$-distributions on the manifolds
%%of dimension $\leq 6$, generic germs of $4$-distributions %%on   the manifolds
%%of dimension $\leq 10$,  generic germs of $5$-%%distributions on   the manifolds
%%of dimension $\leq 15$ are $2$-generating.

The class of generic medium fat distributions is yet bigger.

\newtheorem{gemf}[ge2ge]{Theorem}
\begin{gemf}{\em \small\bf (Generic Medium Fat Distributions)}
\label{gem}
Generic germ of rank $r$  distribution on $n$-dimensional manifold
with $r \leq  n  \leq r+(r-1)^{2}$ is medium fat. $\Box$
\end{gemf}


%%Thus  for example generic germs of
%%$3$-distributions
%%on the manifolds
%%of dimension $\leq 7$, generic germs of $4$-distributions %%on   the manifolds
%%of dimension $\leq 13$,  generic germs of
%%$5$-distributions on   the manifolds
%%of dimension $\leq 21$ are medium fat.


The proof of the latter  result is more technically  involved; it can be found in the Subsection~\ref{gmf}.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proofs}
\label{proo}
\setcounter{equation}{0}


\subsection{Theorem~\ref{wmn} on  Stable  Local Openness: Sketch of the Proof}
\label{losur}

%%\newtheorem{wmne}[nemin]{Theorem}
%%\begin{wmne}%
%%{\em {\small\bf
%%(Morse Index and Stable Local Openness of a Smooth Map)}}
%%\label{wmn}
%%Let $X$ be  a Banach space and
%%$\hx \in X$ be a corank $m$ critical point
%%for the map  $G:X \rightarrow  R^{k}$,
%%which is twice Frechet differentiable at $\hx$.
%%If the Morse index of $\hx$ is $\geq m$, then $G$
%%is stable locally open, i.e. for any neighborhood
%%$U$ of $\hx \in X$ there exists a $C^{0}$-neighborhood
%%${\cal V}$ of $G$ such that for any $\bar{G} \in {\cal V}: \
%%G(\hx) \in \vnu \bar{G}(U). \ \Box$
%%\end{wmne}



We assume without loss of
generality, that $\hx$ coincides with  the origin of $X$.
We denote by $D$ and $h$, correspondingly,
the differential and the Hessian
of the map  $F$ at the origin.
Suppose, that for any
$\lambda \in (\im D)^{\perp}$
index of the quadratic form
$\lambda h$ on $\ker D$
is  $\geq m.$
We are going to prove that then
$F$ is locally open  at $\hx$ and this property is stable, i.e. persists under small perturbations of $F$ in $C^{0}$-metric.


For any $w \in  R^{k}$
the equation $F(x)=w$ can be represented as
a system of equations  $f(y,z)=u, \ g(y,z)=v$, where
$(y,z)=x, (u,v)=w$ are such splittings of $x$ and $w$,
that: i) $u$ coordinatizes $\mbox{Im} D$; ii) $v$ cordinatizes a
complementary space to $\mbox{Im} D$;  \\
iii) $z$ coordinatizes $\ker D, \ \mbox{and} \
 \partial F/\partial z=0$;
$$\mbox{iv)} \ \dim f= \dim y=\dim u=\rk D=k-m, \
\rk \partial F/\partial y |_{0}=\rk \partial f/\partial y |_{0}
=\rk D$$
By  virtue of Implicit Function Theorem
the equation $f(y,z)=u$ can be resolved
w.r.t. $y: \ y=y(z,u).$
The function $y(z,u)$ is continuous locally and $y(0,0)=0$.
Substituting $y(z,u)$ into the equation $g(y,z)=v$ we obtain an
equation $g(y(z,u),z)=0$.
Obviously
local openness  of $F$ at the origin
is equivalent to local  openness  of the map $(u,z) \mapsto (u,g(y(z,u),z))$.
To establish this  local openness  it is   enough to
prove that  the map  $\varphi(z)=g(y(z,0),z)$
covers  the origin of $ R^{k}$  {\em substantially},
i.e. for any closed neighborhood $U$ of
the origin in $X$ there exists $\delta >0$ such that
$$ \|\bar{\varphi} - \varphi\|_{C^{0}(U)} < \delta \Rightarrow
0 \in \vnu \bar{\varphi}(U).$$


We are dealing now with the map
$\varphi$, whose differential vanishes at
the origin: $\varphi'(0)=0$.
To avoid additional notation we will assume,
instead of it, that
$D=F'|_{0}=0$, and then $h=F''|_{0}$ is a quadratic map
of $X$ into $ R^{m}$. Again we assume, that
for any nonzero
$\lambda \in  R^{m^{*}}$
index of the quadratic form
$\lambda h$  is $\geq m.$


First we  get rid of infinite-dimensional space $X$,
 proving that
under the conditions of the Theorem \ref{wmn}
there exists a finite-dimensional subspace
$W \subset X,$
such that for any nonzero
$\lambda \in  R^{m^{*}}$
index of the quadratic form
$\lambda h|_{W}$  is $\geq m$.


Indeed for any unit covector
$\bar{\lambda} \in  R^{m^{*}},$
there exists a $m-$dimensional subspace
$W_{\bar{\lambda}} \subset X,$ such that the restriction
$\bar{\lambda} h|_{W_{\bar{\lambda}}}$ is positive definite.
For all $\lambda$'s
from some small neighborhood $\Omega_{\bar{\lambda}}$ of $\bar{\lambda}$
the quadratic forms $\lambda h|_{W_{\bar{\lambda}}}$ are also
positive definite. Choosing a finite covering of the sphere
$\|\lambda\|=1$ by corresponding neighborhoods
$\Omega_{\bar{\lambda^{1}}}, \ldots \Omega_{\bar{\lambda^{s}}}$
we may take $W=W_{\bar{\lambda}^{1}}+ \cdots W_{\bar{\lambda}^{s}}$.


From now on we consider $W$ in place of $X$
or, all the same, assume $\dim X < \infty$.

The following statement enables us
to study  the quadratic map   $h$ instead of $F$.

\newtheorem{quadr}{Lemma}[subsection]
\begin{quadr}
\label{quad}
If the cone $h^{-1}(0)$ contains a regular point of the
quadratic map  $h:X \rightarrow  R^{m}$,
then $F$ is locally open  in a stable way.  $\Box$
\end{quadr}

{\bf Proof.} If $y \in h^{-1}(0)$
is a regular point of $h$, then there exists a
$m$-dimensional subspace $Z \subset X$, such that
$h|_{y+Z}:(y+Z) \rightarrow  R^{m}$
is local diffeomorphism at $y$. Since $h$ is homogeneous,
then the same holds for all points $\eta y, \ \eta \neq 0$.

Consider the map  $\phi_{\epsilon}(z)=h(y+\epsilon z),$
where $z$ belongs to the unit sphere $S^{m-1} \subset Z$.
Obviously $h(\eta y+ \eta \epsilon z)=\eta^{2}\phi_{\epsilon}(z)$
and, for all sufficiently small  $\epsilon >0$,
the topological
degree of the map
$\phi_{\epsilon} / \|\phi_{\epsilon}\|:S^{m-1} \rightarrow S^{m-1}$
is  $+1$  or  $-1$.
Since the differentials of $h$ at the points $\eta y$ are
nondegenerate, then $\exists a>0$, such that
for small enough $\epsilon >0$  and    $\forall z \in S^{n-1}$:
$\| h(\eta y + \eta \epsilon z) \| \geq a \eta^{2} \epsilon $.
Also for any $w$ with $\|w\| < a \eta^{2} \epsilon$ there exist
$\zeta$ with $\|\zeta\| \leq 1$ such that
$h(\eta y + \eta \epsilon \zeta)=w$.
On the other side
$$\|F(\eta y+\eta \epsilon z)-h(\eta y+\eta \epsilon z)\|=o(\eta^{2})$$
and therefore for some $\epsilon >0$ and small enough $\eta >0$
the topological degree of the map
$$z \ar F(\eta y+\eta \epsilon z)/\| F(\eta y+\eta \epsilon z)\|$$
is +1 or -1. Hence one can  choose  small enough $\eta >0$
such that say for $\|w\| <(\epsilon/2)a \eta^{2}  $
the equation $F(\eta y+\eta \epsilon z)=w$ has a solution $z_{\eta}(w)$
belonging to the unit ball $B \subset Z$.
As far as we apply the topological degree argument
the same will be true for any map $\bar{F}$ which is sufficiently
close to $F$ in $C^{0}$-metric.
$\Box$

This Lemma allows us
to deal with the quadratic map  $h$ instead of $F$.
The conclusion of the Theorem \ref{wmn}
follows from the following


\newtheorem{repreg}[quadr]{Lemma}
\begin{repreg}
\label{rep}
Let $P:X \rightarrow  R^{m}$ be quadratic map,
such that
$\ind \lambda P \geq m, \ \forall \lambda \in  R^{m^{*}}
\setminus {0}.$ Then $P^{-1}(0)$ contains regular point
of the map  $P. \ \Box$
\end{repreg}

Proof of this Lemma  can be found  in
\cite[Appendix 1, Lemma 9]{AS2}
%%Without loss of generality we may assume, that
%%$P'|_{x}=0$ if and only if $x=0.$ Indeed otherwise the condition
%%$P'|_{x}=0$ means, that $x$
%%lies in the intersection of the kernels
%%of the (quadratic forms, which are)
%%components of $P$, and we may take
%%quotient of $X$ w.r.t. to this intersection.
%%
%%Let us start induction w.r.t. $m.$ For $m=1$
%%the theorem was proved in \cite{AS1}.
%%Taking $m>1$ we will treat separately two
%%cases.
%%
%%i) $\underline{P^{-1}(0) \neq \{0\}.}$ Let $P(y)=0$ for some
%%$y \neq 0$. Then $P'|_{y} \neq 0$
%%and $\lambda P''|_{y}$ coincides with the restriction of
%%$\lambda P$ onto $\ker P'|_{y} \
%%(\lambda \perp \im  P'|_{y})$.
%%Conditions
%%of the Proposition imply
%%$\ind \lambda P''|_{y} \geq \codim \im P'|_{y}, \
%%\forall \lambda \perp \im  P'|_{y}.$
%% Since $\dim \mbox{coker}P'|_{y} <m,$ then according
%%to the inductive assumption the inverse image
%%$(P''|_{y})^{-1}(0)$ contains regular point
%%and hence $P^{-1}(0)$ contains regular point.
%%
%%ii) $\underline{P^{-1}(0)=\{0\}.}$
%%Then $\im P$ is a {\em closed} cone.
%%Applying inductive assumption to
%%the Hessians $P''|_{y}$ for any $y \neq 0$ we obtain,
%%that $\forall z \in \im P \setminus {0}$
%%the inverse image $P^{-1}(z)$ contains  regular point of $P$
%%and therefore $\im P \setminus {0}$ is {\em open}.
%%This means that $\im P= R^{m}.$

%%Let $S$ be unit sphere in $X$. Then
%%$P/\|P\||_{S}:S \rightarrow S^{m-1}$ is a surjective map.
%%In virtue of Sard theorem there exists
%%regular value $v$ of this map. If $P(x)=av$ for some $a >0,$
%%then $\rk (P|_{S})'_{x} \geq m-1.$
%%Let $a_{v}=\min\{a>0|av \in P(S)\}; \ a_{v}>0,$
%% since $P^{-1}(0)={0}$.
%%Let $x_{v} \in S$ and $P(x_{v})=a_{v}v$.
%%The pair $(a_{v},x_{v})$ is point of local minimum and
%%normal extremal point for the following problem
%%of relative extremum:
%%$$a \rightarrow \min, \ P(x)-av=0.$$
%%Standard 2nd-order necessary optimality condition for this
%%problem provides
%%existence of $\lambda \in  R^{m*}} \setminus 0$, such
%%that:
%%$$\lambda v \geq0, \ \lambda(P|_{S})'|_{x_{v}}=0,
%%\ \lambda (P|_{S})''|_{x_{v}} \geq 0.$$

%%Direct computation gives
%%$$1/2\lambda (P|_{S})''|_{x_{v}}(y)=
%%\lambda P(y)-|y|^{2}\lambda P(x_{v}),$$
%%and hence $\lambda P(y) \geq 0$ for all
%%$y \in N=\{y \mid y \perp x_{v}, P'|_{x_{v}}y=0\}$.
%%Obviously $N$ is a linear subspace of codimension
%%$m$ in $X$. Since $\lambda P(x_{v})=\lambda v \geq 0$
%%and $x_{v}$ is orthogonal and $P-$orthogonal to $N$ ,
%%then $\lambda P$ is nonnegative on $N \oplus \sp\{x_{v}\}$
%%and hence $\ind \lambda P \leq m-1$, i.e.
%%we come to a contradiction, which
%%{\bf finishes the proofs of the Proposition \ref{rep} and
%%Theorem \ref{lagne}.} $\ \bbox$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\subsection{Proof of the Theorem~\ref{compa}  on compactness
of the set of minimal geodesics of given length}
\label{compset}
 Assume all length-minimizers,
we consider,
to be of  positive length $\rho <1$.
 It will be  convenient for us  to parameterize
them  by the  length of arc. We assume that the vector fields
$g^{i}(q), \ i=1, \ldots , r$,
form an  orthonormal basis in
$T_{q}M$ for all $q$ from a small neighborhood of $q^{0}$.
Hence the corresponding minimizing controls
have constant magnitude $1$.

As we already know   all  length-minimizers  for medium fat
 sub-Riemannian structures  are   normal geodesic paths.
(The proof of this fact can be found  in the next subsection.)
%%%it shares some technique -
%%%estimates for Morse index - %%%with the present proof, %%%but is completely independent %%%from  it).
 We assume $\hp_{0} =-1/2$ and
 consider (multiplied by~$2$)
second variation along a fixed normal geodesic path
 of length $\rho$.   This
is a quadratic form (compare with (\ref{4.2})-(\ref{dom2})):
  \begin{equation}                                           \label{C.1}
Q(u(\cdot))=
-\int_{0}^{\rho}\langle u(\tau), u(\tau) \rangle d \tau +
 \hp_{T} \int_{0}^{\rho}
\left[\int_{0}^{\tau}Y_{\xi}u(\xi)d\xi,
Y_{\tau}u(\tau)\right](q^{0})d\tau ,
\end{equation}
defined on the set of $u(\cdot)$'s which satisfy
\begin{equation}                           \label{C.2}
\int_{0}^{\rho}\hu(\tau) u(\tau) d \tau =0, \
   \int_{0}^{\rho} Y_{\tau}(q^{0})u(\tau) d \tau =0.
\end{equation}
Recall that $\hp_{T}$ coincides with the initial value $\hp(0)$ of the adjoint covector function $\hp(t)$ and
$$Y_{\tau}=(Y^{1}_{\tau} \ldots Y^{r}_{\tau}), \ Y^{i}_{\tau}=\Ad  \chro_{0}^{\tau} G \hu(\xi)d\xi g^{i}, \
i=1, \ldots , r. $$
Due to real analyticity of any normal extremal control $\hu(\cdot)$
the  vector fields $Y^{i}_{\tau}$ are real analytic.
For any small compact
neighborhood $K$ of $q^{0}$ their
$C^{1}$-norms $\|Y^{i}_{\tau}\|_{1,K}$   have a  uniform
(with respect
to $\hu(\cdot)$ and $\tau  \in [0,\rho]$)   upper estimate $B$.




%%By means of the Riemannian inner product in $T_{q}M$
%%we  identify
%%$T_{q^{0}}M$ with $T^{*}_{q}M$ and
%%may represent   $\hp_{T}$  as  a sum of
%%vertical component   $\hp_{T}^{v}$, which is
%%orthogonal to the distribution ${\cal D}_{q}$ and the   %%'horizontal'
%%component $\hp_{T}^{h}=\hp_{T}-\hp_{T}^{v}$:
%%$$\hp_{T}^{h}=\hp_{T}G(q^{0})=
%%\sum_{i=1}^{r}\langle \hp_{T},g^{i}(q^{0})\rangle %%g^{i}(q^{0}), \  \hp_{T}^{v}=\hp_{T}-%%\sum_{i=1}^{r}\langle \hp_{T},g^{i}(q^{0})\rangle %%g^{i}(q^{0}).$$

%%Obviously $\hp_{T}^{h} \neq 0$.
%% As long as
%%$u(\tau)=\hp(\tau)G(q(\tau))$ and $\|u(\tau)\|=1$, the
%%$|\hp^{h}(\tau)G(q(\tau))=1$.
%%\marginpar{normirovka!}

Our first goal is establishing  that for given $\rho>0$
\begin{center}
{\em $\exists B$  such that if  $\|\hp_{T}\| \geq B$, then   the corresponding geodesic path
$\|\hp_{T}\| \geq B$  fails to be  a length-minimizer.}
\end{center}
We shall also estimate the asymptotic dependence of $B$ on $\rho$ as
$\rho \rightarrow +0$.


An easy part of the proof amounts to  excluding 'too fast' oscillating geodesic controls from the candidates to length-minimizers. The harder part of the proof is establishing that, if the geodesic control is not too fast oscillating,  then both positive and negative indices of the second variation   become arbitrarily  large  provided that   $\|\hp_{T}\|$ is sufficiently large . Therefore according to the Theorem~\ref{wmn} the geodesics which correspond to large $\|\hp_{T}\|$ fail to be length-minimizer
{\small\bf  Remark.}  When establishing
the above mentioned result about indices one may ignore the conditions (\ref{C.2}) , as far as their imposition diminishes the value of indices at most by $n+1 \ (n=\dim M)$.


To evaluate the rate of oscillation we introduce  the following
norm, which  is useful for studying fast oscillating and relaxed controls.
\newtheorem{recon}{Lemma}[subsection]
\begin{recon}
\label{reco}
The expression
$$\|u(\cdot)\|_{o}=\sup_{t',t'' \in [0,\rho]}
\|\int_{t'}^{t''}u(\tau)d \tau\|$$
defines a norm in the space of integrable functions (the corresponding topology is obviously weaker then the one of $L^{r}_{1}[0,\rho]). \  \Box$
\end{recon}

(The subindex 'o' stays for 'oscillating'.) One can find in \cite{Gam} examples of using this norm, e.g. a theorem on continuous,  with respect to this norm, dependence of solutions of ODE on right-hand side.

The following technical result is local  and we    provide  its formulation
 for a germ at $0_{ R^{n}}$
of distribution ${\cal D}=\mbox{span}\{G(x)u \mid u \in  R^{r}\}$ in $ R^{n}$  in place  of  $M$.

\newtheorem{condep}[recon]{Lemma}
\begin{condep}(compare with \cite[Chap. 4]{Gam})
\label{conde}
For any  sufficiently small $\rho_{0}>0$ there exists a constant $C>0$, such that for any $\rho < \rho_{0}$ and
for all  $u(\cdot) \in L_{\infty}^{r}[0, \rho]$
such that $\|u(\cdot)\|_{L_{\infty}} \leq 1$,
the corresponding trajectories
of the distribution satisfy
\begin{equation}                     \label{gu}
\dot{x}(t)=G(x(t))u(t), \ x(0)=0 \  \Rightarrow
x(\rho) \leq C\|u(\cdot)\|_{o}. \ \Box
\end{equation}
\end{condep}

It has been already mentioned in the Subsection~\ref{distr}
that for a  sub-Riemannian structure determined by a
distribution of degree of nonholonomy $\kappa$
the Riemannian distance  from the center $q^{0}$
to the sub-Riemannian $\rho$-sphere admits (for $\rho < \rho_{0}$)
a lower estimate $\ul{c}\rho^{\kappa}$ with $\ul{c}>0$ depending continuously
on $q^{0}$.


Therefore coming back to the germ of distribution $\cal D$ at
 $q^{0} \in M$  of degree of nonholonomy $\kappa$ we
conclude with the following

\newtheorem{nonos}[recon]{Corollary}
\begin{nonos}
\label{nono}
For some $\rho_{0}>0$ there exists a constant $c>0$
such that $\forall \rho < \rho_{0}$
and for any control $u(\cdot)$
leading from $q^{0}$ to a point of sub-Riemannian
$\rho$-sphere there  holds
$\|u(\cdot)\|_{o} \geq c\rho^{\kappa}. \Box$

\end{nonos}

{\em Proof of the Lemma}~\ref{conde}. Let $\|G(0)\|=L_{0}$ and $L \geq 0$ be  a Lipschitz constant for the restriction
of $G(x)$  on  some compact neighborhood of $0$.
If  $x(t)$  stays inside this  compact neighborhood then
 transforming  the differential equation
(\ref{gu})  into an integral one:
\begin{equation}                     \label{inteq}
x(t)=\int_{0}^{t} G(x(\tau))u(\tau)d \tau
\end{equation}
we conclude
$$\|x(t)\| \leq \int_{0}^{t} \|G(x(\tau))u(\tau)\|d \tau
\leq  \int_{0}^{t} (L \|x(\tau)\| +L_{0})d\tau,$$
and using the Gronwall inequality  obtain
$$\|x(t)\| \leq L_{0}L^{-1}(e^{Lt}-1), \
\mbox{\rm if} \ L>0,
\ \mbox{\rm and} \ \|x(t)\| \leq L_{0}t, \ \mbox{if} \ L=0.$$
Proceeding with the  integration   by parts in (\ref{inteq})
we obtain
$$x(t)=G(x(t))\int_{0}^{t}u(\tau)d \tau -
  \int_{0}^{t}\left((G(x(\tau))u(\tau)) \circ G\right)(x(\tau))
  \int_{0}^{\tau}u(\xi)d\xi d\tau .$$
 As long as  $\sup_{\tau}| \int_{0}^{\tau}u(\xi)d\xi| \leq \|u(\cdot)\|_{o}$ and $\|u(\cdot)\|_{L_{\infty}} \leq 1$,
then $\|x(t)\| \leq C\|u(\cdot)\|_{o}$, where the constant
$C$ is calculated via $\|G(\cdot)\|_{C^{1}}. \Box$






Let us  introduce the covector-functions $\psi^{1}(t),\psi^{2}(t)$ with the components:
\begin{eqnarray}
\psi^{1}_{i}(t)=\hp(t)g^{i}(\hq(t)), \  i=1, \ldots ,r; \label{psi1} \nonumber \\
\psi^{2}_{ij}(t)=\hp(t)[g^{i},g^{j}](\hq(t)),
                                 \  i,j=1, \ldots ,r.   \label{psi2}
\end{eqnarray}

According to the Pontryagin Maximum Principle
(Theorem~\ref{pont}) $\hu(t)=\hp(t)G(\hq(t))=\psi^{1}(t)$ and hence
$\|\psi^{1}(t)\|=\|\hu(t)\| \equiv 1.$
Also
$\psi^{2}_{ij}(t)=\hp_{T}[Y^{i}_{t},Y^{j}_{t}](q^{0})$.




%%Representing the
%%second variation in the form
%%\begin{eqnarray}                                   %%\label{C.3}
%%\int_{0}^{\rho}\langle u(\tau), u(\tau) \rangle d \tau +
%%\hp_{T}^{h}
%%\int_{0}^{\rho}[\int_{0}^{\tau}Y_{\xi}u(\xi)d\xi,Y_{\tau}%%u(\tau)](q^{0})d\tau
%%\nonumber \\
%%+\hp_{T}^{v}
%%\int_{0}^{\rho}[\int_{0}^{\tau}Y_{\xi}u(\xi)d\xi,Y_{\tau}%%u(\tau)](q^{0})d\tau,
%%\end{eqnarray}
%%let us notice that the sum of the first two summands %%admits  a uniform
%%  upper estimate
%%$A \rho \|u(\cdot)\|_{L_{2}}^{2}$, with  $A>0$ neither %%depending
%%on $\hp_{T}$ nor on a choice of geodesic control %%$\hu(\cdot)$.

%%Now we are going to derive a lower estimate
%%for the last addend in (\ref{C.3}).




The rest of our proof is based on the following technical result.



\newtheorem{lowe}[recon]{Lemma}
\begin{lowe}
\label{lwe}
i) If the distribution $\cal D$ is $2$-generating at
a point $q^{0} \in M$,
then there exist  constants   $\delta \in (0,1/2), A>0,
\rho_{0}>0, C>0$
such that for all geodesics  of length
$\rho \leq \rho_{0}$:
$$
%%\inf_{\psi \in {\cal D}_{q}^{\perp}, \ \|\psi\|=1}
%%\max_{1 \leq i,j \leq r}
%%|\psi[Y^{i}_{0},Y^{j}_{0}](q)| =|\psi[g^{i},g^{j}](q)|
\|\hp_{T}\| \geq A \Rightarrow \|\psi^{2}(t_{0})\| \geq C \delta \|\hp_{T}\|$$
at some point $t_{0}$  of the interval $[0, \rho]$;

 ii) if the distribution $\cal D$ is medium fat
at a point $q \in M$,
then there exist  constants  $\delta \in (0,1/2), A>0,\rho_{0}>0$ such that  for all  geodesics
$(\hu(\cdot)|_{[0,\rho]}, \hq(\cdot)|_{[0,\rho]},\psi_{0},\hp(\cdot)|_{[0,\rho]})$,
whose end-point $\hq(\rho)$ belongs  to  the $\rho$-sphere
($0 < \rho < \rho_0$),   there holds:
\begin{equation}
\label{estmf}
\|\hp_{T}\| \geq A \Rightarrow \|\psi^{2}(t_{0})\| \geq  C \delta \rho^{3} \|\hp_{T}\|,
\end{equation}
at some point  $t_{0}$  of the interval $[0, \rho]$. $\Box$
\end{lowe}
{\em Remark.} In both statements the point $t_{0}$ may depend on  geodesic, but the constants $k, \delta , \rho_{0}, C$ can be chosen in a uniform way.

{\em Proof of the Lemma~\ref{lwe}}.
Proof of i) is immediate. Indeed, if
$\cal D$ is  2-generating,  then  the vector fields
$g^{i}, [g^{i},g^{j}], \ i,j \in \{1, \ldots ,r\}$,
span the space $T_{q}M$ and therefore
$\|\hp(t)\| \leq \mu (\|\psi^{1}(t)\|+\|\psi^{2}(t)\|)$.
Then
$$A \leq  \|\hp_{T}\|=\|\hp(0)\| \leq \mu(\|\psi^{1}(0)\|+ \|\psi^{2}(0)\|), $$
and
$$\|\psi^{2}(0)\|  \geq  \mu^{-1}\|\hp_{T}\|-1 \geq
(\mu^{-1}-1/A)\|\hp_{T}\|.$$



{\em Proof of ii).} Recall that we deal with  a geodesic of positive length $\rho \leq \rho_{0}$ (some conditions on
 $\rho_{0}$ will be imposed later).
Introduce the covector-function  $\psi^{3}(t)$:
$$\psi^{3}_{ijk}(t)=\hp(t)[g^{i},[g^{j},g^{k}]](\hq(t)),
                                  i,j,k=1, \ldots ,r.$$

As long as the degree of nonholonomy of medium fat sub-Riemannian  distributions is $\leq 3$, then
$$\|\hp(t)\| \leq \mu  (\|\psi^{1}(t)\|+\|\psi^{2}(t)\|+\|\psi^{3}(t)\|).$$

Differentiating the functions $\psi^{1}(t),\psi^{2}(t),\psi^{3}(t)$  with respect to $t$ we  obtain

\begin{eqnarray}
\dot{\psi}^{1}_{i}(t)=\hp(t)[G\hu(t),g^{i}](\hq(t)),  i=1, \ldots ,r;  \nonumber \\
\dot{\psi}^{2}_{ij}(t)=
\hp(t)[G\hu(t),[g^{i},g^{j}]](\hq(t)),
                                  i,j=1, \ldots ,r;  \label{dif2} \\
\dot{\psi}^{3}_{ijk}(t)=
\hp(t)[G\hu(t),[g^{i},[g^{j},g^{k}]]](\hq(t)),
                                  i,j,k=1, \ldots ,r.
\nonumber
\end{eqnarray}
Using the notation $\psi(t)$ for the triple
$(\psi^{1}(t),\psi^{2}(t),\psi^{3}(t))$ we
conclude that for geodesics of length $\rho < \rho_{0}$ there exists a constant $B>0$ depending only on the distribution ${\cal D}$ (and on $\rho_{0}$) such that
$$\|\dot{\psi}(t)\| \leq B\|\psi(t)\|.$$

Using the Gronwall inequality we
 obtain  $\forall \bar{t} \geq
\ul{t}$:
\begin{equation}                          \label{gr1}
\|\psi(\bar{t})\| \leq  \|\psi(\ul{t})\|e^{B(\bar{t}-\ul{t})},
\end{equation}
and also
\begin{eqnarray}
\|\psi(\bar{t})- \psi(\ul{t})\| \leq
\| \psi(\ul{t})\| \int_{\ul{t}}^{\bar{t}}
Be^{B(\tau-\ul{t})}d \tau =
\| \psi(\ul{t})\|(e^{B(\bar{t}-\ul{t})}-1)
,  \nonumber \\
\label{gr2}
\|\psi(\bar{t})\|\geq \|\psi(\ul{t})\|(1-(e^{B(\bar{t}-\ul{t})}-1)).
\end{eqnarray}
If $\rho >0$ is fixed and $t,t_{0} \in [0, \rho]$, then we  may
derive from  the  inequalities (\ref{gr1})-(\ref{gr2})
 (increasing $B$ if necessary) the estimate:
 $$\|\psi(t)\|\geq \|\psi(t_{0})\|(1-B|t-t_{0}|).$$
In a similar way we derive $\forall \bar{t} \geq
\ul{t}$:
\begin{eqnarray}
\|\psi_{2}(\bar{t})- \psi_{2}(\ul{t})\| \leq \| \psi(\bar{t})\|B(\bar{t}-\ul{t}),  \label{raz2} \\
\|\psi_{2}(\bar{t})\| \geq  \|\psi_{2}(\ul{t})\|-  \|\psi(\ul{t})\|B(\bar{t}-\ul{t}))   \label{niz2}
\end{eqnarray}
with (probably  different)  constant $B$ depending  on
$\cal D$ but neither  depending on  geodesic nor on $\rho < \rho_{0}$.
In what follows we assume $B\rho_{0} <1$.



%%If  $\|\psi(0)\| \geq k$ then
%%$\forall t \in [0,T]: \
%%\|\psi(t)\|\geq \geq k(1-B\rho).$

As far as  medium fat distribution has degree  of nonholonomy
(at most) $3$, then
according  to the Corollary~\ref{nono}
any geodesic control $\hu(\cdot)$,  which drives us from $q^{0}$ to a point of $\rho$-sphere, must   satisfy  the condition
$\|\hu(\cdot)\|_{o} \geq c\rho^{3}$, that means that for some
 $t_{0},  t_{1} \in [0, \rho ]: \
\|\int_{t_{0}}^{t_{1}}\hu(\tau)d\tau\| \geq c\rho^{3}$.
Obviously $t_{1}- t_{0} > c\rho^{3}$.

Coming back to the proof of the lemma~\ref{lwe} let us choose some $\delta \in (0,1/2)$; some additional restrictions on $\delta$
will  be imposed later.

Take  the point $t_{0}$.  If
$$ \ \|\psi^{2}(t_{0})\| \geq
B\delta \rho^{3} \|\psi(t_{0})\| \geq \mu^{-1}(1-B\rho_{0})B\delta \rho^{3} \|\psi_{T}\|,$$
then the statement ii) of the lemma is proved.
%%Then
%%$$ \|\psi^{2}(t)\| \geq \|\psi^{2}(t_{0})\| -
%% \|\psi(t_{0})\|B|t-t_{0}|
%%\geq B(\delta \rho - |t-t_{0}|).$$
%%Hence $\|\psi^{2}(t)\| \geq B\delta \rho/2$ on both of %%the intervals $[t_{0}, t_{0}+\delta \rho/2],
%%[t_{0}-\delta \rho/2, t_{0}]$.

Let now
$\|\psi^{2}(t_{0})\| \leq
B\delta \rho^{3} \|\psi(t_{0})\|$.
Without loss of generality one may assume (provided
 $A$ in (\ref{estmf}) is big enough)
$$\|\psi^{3}(t_{0})\| \geq 2^{-1} \|\psi(t_{0})\|.$$

From the second one
of the differential equations  (\ref{dif2})
we derive
\begin{eqnarray}
\hspace*{-4mm} \psi^{2}_{ij}(t_{1})- \psi^{2}_{ij}(t_{0})=
\int_{t_{0}}^{t_{1}}
\hp(\tau)\left[G\hu(\tau),[g^{i},g^{j}]\right](\hq(\tau))
d\tau=
\label{dpsi2} \\
\hp(t_{0})\left[G\int_{t_{0}}^{t_{1}} \! \! \hu(\tau)d \tau ,
[g^{i},g^{j}]\right](\hq(t_{0})) \! \! + \!
\int_{t_{0}}^{t_{1}} \! \!
\hp(\tau)\left[G\hu(\tau),[G\int_{\tau}^{t_{1}}\! \! \hu(\xi)d \xi ,
[g^{i},g^{j}]]\right](\hq(\tau))d\tau .
\nonumber
\end{eqnarray}

 As far as the distribution is medium fat then for some
$i_{0},j_{0} \in \{1, \ldots , r\}$ there holds
$$ \hp(t_{0})[G\int_{t_{0}}^{t_{1}}\hu(\tau)d \tau ,
[g^{i_{0}},g^{j_{0}}]](\hq(t_{0})) \geq b \|\psi^{3}(t_{0})\|
 \|\int_{t_{0}}^{t_{1}}\hu(\tau)d \tau \| \geq
2^{-1}b \|\psi(t_{0})\|
 \|\int_{t_{0}}^{t_{1}}\hu(\tau)d \tau \|.$$
For any $i,j$
the norm of the second addend
at the right-hand side  of (\ref{dpsi2})
admits an upper estimate
$C\rho e^{B\rho}\|\psi(t_{0})\|
 \|\int_{t_{0}}^{t_{1}}\hu(\tau)d \tau \|$. Without loss of generality we may assume $C\rho_{0} e^{B\rho_{0}} < 2^{-2}b$ thus coming to the
lower estimate
$$|\psi^{2}_{i_{0}j_{0}}(t_{1})- \psi^{2}_{i_{0}j_{0}}(t_{0})|
\geq 2^{-2}b \|\psi(t_{0})\|
 \|\int_{t_{0}}^{t_{1}}\hu(\tau)d \tau \| \geq 2^{-2}bc  \|\psi(t_{0})\|
\rho^{3}.$$



Now recalling that  $\|\psi^{2}(t_{0})\| \leq
B\delta \rho^{3} \|\psi(t_{0})\|$   we obtain
$$|\psi^{2}_{i_{0}j_{0}}(t_{1})| \geq -B\delta \rho^{3}
\|\psi(t_{0})\|+2^{-2}bc\rho^{3}\|\psi(t_{0})\|.$$
Choosing $\delta < bc/8B$ we  conclude
$$\psi^{2}_{i_{0}j_{0}}(t_{1}) \geq
2^{-3}bc\rho^{3}\|\psi(t_{0})\| \geq
 B\delta \rho^{3}\|\psi(t_{0})\| \geq
B \delta \rho^{3}(1-B\rho_{0})\mu^{-1}\|\hp_{T}\|. \Box$$


Now we may complete  the proof of the Theorem.

Integrating the second addend of (\ref{C.1}) by parts we transform the second variation into

\begin{eqnarray*}
-\int_{0}^{\rho}\langle u(\tau), u(\tau) \rangle d \tau +
 \hp_{T} \int_{0}^{\rho}
[Y_{\tau}\int_{0}^{\tau}u(\xi)d\xi,
Y_{\tau}u(\tau)](q^{0})d\tau -         \\
  \hp_{T} \int_{0}^{\rho}
[\int_{0}^{\tau}\dot{Y}_{\xi}\int_{0}^{\xi}u(\theta)
d\theta d \xi,
Y_{\tau}u(\tau)](q^{0})d\tau.
\end{eqnarray*}

Let us take the point $t_{0}$ whose existence has been  established  by  the  Lemma~\ref{lwe}.
Take a subinterval
of  length $\delta \rho^{\alpha}$ joining (for example beginning at) this point.  Put   $\alpha=1$,  when  dealing  with a $2$-generating distribution, and
choose any $\alpha >3$,  when
dealing  with a medium fat distribution.

We restrict our consideration to
controls $u(\cdot)$
of magnitudes $\leq 1$ with  supports in
the  subinterval $[t_{0},t_{0}+\delta \rho^{\alpha}]$.
Then the second variation  takes form
\begin{eqnarray}    \label{dvalf}
-\int_{t_{0}}^{t_{0}+\delta\rho^{\alpha}}\langle u(\tau), u(\tau) \rangle d \tau +
 \hp_{T} \int_{t_{0}}^{t_{0}+\delta\rho^{\alpha}}
[Y_{\tau}\int_{t_{0}}^{\tau}u(\xi)d\xi,
Y_{\tau}u(\tau)](q^{0})d\tau -         \nonumber \\
  \hp_{T} \int_{t_{0}}^{t_{0}+\delta\rho^{\alpha}}
[\int_{t_{0}}^{\tau}\dot{Y}_{\xi}
\int_{t_{0}}^{\xi}u(\theta)
d\theta d \xi,
Y_{\tau}u(\tau)](q^{0})d\tau.
\end{eqnarray}
Let us note that for fixed $\hp_{T}$  and $\rho \rightarrow +0$ each integration in (\ref{dvalf})  adds a factor  $\rho^{\alpha}$
to the order of smallness.


The norm of the    first addend admits  an upper \footnote{We can choose common constant $\bar{C} >0$ in this and the subsequent upper estimates.
This constant does not depend on geodesic} estimate  $\bar{C}\rho^{\alpha}$.
 The last addend admits an upper estimate $\bar{C}\|\hp_{T}\|\rho^{3\alpha}$.
According to the formula (\ref{raz2})
$$\forall t \in [t_{0},t_{0}+\delta \rho^{\alpha}]: \
\left|\hp_{T}[Y_{t}v,Y_{t}u](q^{0}) -\hp_{T}[Y_{t_{0}}v,Y_{t_{0}}u](q^{0})\right| \leq
\bar{C}\|\hp_{T}\|\rho^{\alpha}$$
and hence the second addend in (\ref{dvalf})
can be represented as
$$\hp_{T} \int_{t_{0}}^{t_{0}+\delta\rho^{\alpha}}
[Y_{t_{0}}\int_{t_{0}}^{\tau}u(\xi)d\xi,
Y_{t_{0}}u(\tau)](q^{0})d\tau + O(\|\hp_{T}\|\rho^{3\alpha}),$$
where again
$O(\|\hp_{T}\|\rho^{3\alpha}) \leq \bar{C}\|\hp_{T}\|\rho^{3\alpha}$.

Here  $\hp_{T}[Y^{i}_{t_{0}},Y^{j}_{t_{0}}](q^{0})$
are  the components of the covector   $\psi^{2}(t_{0})$. According to the conclusion of the Lemma~\ref{lwe},  $\|\psi^{2}(t_{0})\|$ can be estimated from below by
$C\delta \|\hp_{T}\|$  for a 2-generating distribution, or, by
$C\delta \rho^{3} \|\hp_{T}\|$
for a  medium fat distribution.

Let us consider the quadratic form
\begin{equation}                                   \label{0.9}
G(u(\cdot))=\hp_{T} \int_{t_{0}}^{t_{0}+\delta\rho^{\alpha}}
[Y_{t_{0}}\int_{t_{0}}^{\tau}u(\xi)d\xi,
Y_{t_{0}}u(\tau)](q^{0})d\tau .
\end{equation}

The values of this  quadratic form
depend only on
$\psi_{2}(t_{0})$ (linearly)
and on the length
of the interval.
Introducing a function
$w(\theta)=u(t_{0}+\theta\delta\rho^{\alpha}), \ \theta \in [0,1],$
we arrive to the equality
$$G(u(\cdot))=\delta^{2}\rho^{2\alpha}\hp_{T} \int_{0}^{1}
[Y_{t_{0}}\int_{0}^{\tau}w(\xi)d\xi,
Y_{t_{0}}w(\tau)](q^{0})d\tau.$$



The latter quadratic form possesses
 a symmetry: the involution
$J: w(\theta) \mapsto w(1-\theta)$
changes the signs of this quadratic form.
This implies that for any subspace
on which the form is positive definite
there exists a subspace of the same dimension
on which the form is negative definite.
As long as the kernel of this form has  infinite  codimension
both of those  dimensions  must be infinite.



It follows
from the  aforesaid
(see also \cite[Appendix 2]{AS2}), that
\begin{itemize}
\item in the medium fat   case  for each integer $N>0$ there exist $N$-dimensional subspaces $L^{+}_{N}$ (respectively $L^{-}_{N}$) and constants $A_{N}>0$ such that
 the quadratic form (\ref{0.9})
%%\begin{equation}                  \label{prit}
%%\hp_{T} \int_{t_{0}}^{t_{0}+\delta\rho^{\alpha}}
%%[Y_{t_{0}}\int_{t_{0}}^{\tau}u(\xi)d\xi,
%%Y_{t_{0}}u(\tau)](q^{0})d\tau,
%%\end{equation}
is positive (respectively negative) definite on
$L^{+}_{N}$ (respectively on $L^{-}_{N}$)
and admit a lower  estimate
$b_{N}=A_{N} \rho^{2\alpha+3}\|\hp_{T}\|\|u(\cdot)\|^{2}_{L{\infty}}$
(respectively an upper estimate $-b_{N}$);
\item in the 2
-generating case these estimates change to
$\pm A_{N} \rho^{2}\|\hp_{T}\|\|u(\cdot)\|^{2}_{L{\infty}}$.
\end{itemize}

Therefore if $\|\hp_{T}\| \rightarrow + \infty$  then the values of (\ref{0.9})
grow at least
as $\|\hp_{T}\|\rho^{2\alpha+3}$ or, respectively
in the 2-generating case,
as $\|\hp_{T}\|\rho^{2}$.
Hence we may choose  $\rho_{0}$ in such a way that this principal term  dominates  the terms which grow at most as $\|\hp_{T}\|\rho^{3\alpha}$, as $\rho \rightarrow +0$.     Also  for sufficiently large  $\|\hp_{T}\|$
(larger than $c\rho^{-\alpha -3}$ in the medium fat case, or larger than $c\rho^{-1}$ in the 2-generating case) this principal term will   dominate  the addend
$-\int_{t_{0}}^{t_{0}+\delta\rho^{\alpha}}\langle u(\tau), u(\tau) \rangle d \tau$ which is  $O(\rho^{\alpha})$. In this case  the second variation will have
large  positive and negative indices and hence corresponding geodesic path   would  not be length minimizer.

Therefore for length minimizers of 2-generating sub-Riemannian structures  there must hold  $\|\hp_{T}\|=O(\rho^{-1})$ as $\rho \rightarrow 0$, while for medium fat distribution there must hold
$\|\hp_{T}\|=O(\rho^{-\alpha-3})$, as  $\rho \rightarrow 0$
for any $\alpha >3$.

To obtain the respective results for the geodesics parameterized by the interval $[0,1]$ instead of length we only need to introduce
a time substitution $t \mapsto \rho^{-1}t$ which results in the transformation of other variables $u \mapsto \rho u, \
\psi \mapsto \rho \psi$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%

\subsection{Proof of the Theorem~\ref{mefa}   on Lack of
Strictly Abnormal Minimizers for Medium Fat Distributions}
\label{noab}

Let us consider an abnormal geodesic $\abex$ and the absolutely continuous
covector-function $\psi^{2}(t)$ introduced
by (\ref{psi2}). It can not vanish identically.  Indeed
if $\psi^{2}(t_{0})=0$, and in addition $t_0$ is a Lebesgue point of the function  $t \mapsto \hu(t)$ then  according to
(\ref{dif2})  the components of the derivative
$\dot{\psi}^{2}(t_0)$    equal $\hp(t_{0})[G\hu(t_{0}),[g^{i},g^{j}]](\hq(t_{0}))$.
Due to the medium fatness of the distribution this  derivative
is non vanishing. Therefore
$\|\psi^{2}(t)\| \neq 0$ on some interval
$[t_{0},t_{0}+\nu]$, i.e.  Goh condition fails for the
abnormal extremal. Then according
to the Theorem~\ref{legom}
this  strictly
abnormal  geodesic path  $\pex$ can not be a length minimizer.
$\Box$






\subsection{Proof of the Theorem  on Generic Medium Fat
Distributions}
\label{gmf}



For a  generic germ of $r$-distribution
we may assume  $\dim \cD^{2}_{q}=\frac{r(r+1)}{2}$
(see Section 6, the proof of the proposition~\ref{gem}).

Let $f^{1}, \ldots , f^{r}$ be a basis of the distribution.
Let us enumerate the pairs
$(i,j)$ with   $1 \leq i < j \leq r$,  by means  of one-to-one  map
$n: (i,j) \mapsto n_{ij}, \  n_{ij} \in \{1, \ldots , r(r-1)/2\}$.
Now we introduce a $(\frac{r(r-1)}{2} \times r)$-matrix
$C=(c_{ks}): \  c_{ks}=[f^{s}[f^{i},f^{j}]]$, where  $(i,j)=n^{-1}(k)$.

Obviously a germ of distribution fails to be medium fat
if and only if for some covector
$\psi$ from the annihilator of  $\cD^{2}$  in $T^{*}_{q}M$   the matrix
$C^{\psi}=(c^{\psi}_{ks})=(\langle \psi ,[f^{s}[f^{i},f^{j}]] \rangle)$
has   rank $<r$. Indeed in this case there exists a nonzero vector
$v=(v_{1}, \ldots ,v_{r})$ such that $C^{\psi}v=0$ what  means that
for $X=\sum_{i=1}^{r}f^{i}v_{i}$   the covector $\psi$ annihilates
also $[X,\cD^{2}](q)$.

The  codimension of the (maximal stratum of the stratified) manifold
$\cal S$  of $(\frac{r(r-1)}{2} \times r)$-matrices
of rank $< r$
equals
$\frac{r(r-1)}{2}-(r-1)=\frac{(r-2)(r-1)}{2}$. The codimension of
$\cD^{2}_q$ in $T_{q}M$  is
$\leq r+(r-1)^{2}-\frac{r(r+1)}{2}=\frac{(r-2)(r-1)}{2}$.
These codimensions coincide and were the entries
of the matrix $C$ independent, then we could say that for a generic distribution
$\cD$ there is no nonzero $\psi \in (\cD^{2}_{q})^{\perp}$
for which the corresponding matrix $C^{\psi}$ has rank $<r$
(due to homogeneity we may think that $\psi$ belongs
to the unit sphere in $(\cD^{2}_{q})^{\perp}$).

Clearly the 3rd-order iterated
Lie brackets which appear in
the matrix $C$ are {\em  not}
independent, since for example they  must satisfy   the
 Jacobi identities which will implicate the equality
relations  for the elements of each $C^{\psi}$. This means that the
range of the map
$\psi \mapsto C^{\psi}$ is not the whole space
$ R^{\frac{r(r-1)}{2} \times r}$ but  some linear subspace
$L$ of this space.
This change would  not affect our reasoning if we could  prove
that the codimension of ${\cal S} \bigcap L$ in $L$ coincides
with the codimension of $\cal S$ in $ R^{\frac{r(r-1)}{2} \times r}$.
To do this it is enough to verify that
$L$ is transversal to $\cal S$ in $ R^{\frac{r(r-1)}{2}  \times r}$.

To establish the transversality we will have to describe the relations
between the elements of the matrices $C^{\psi}$
(the relations which determine $L$).
To  proceed with  it we consider   a
Ph.Hall basis (see \cite{Se}) in the free Lie algebra
generated by $f^{1}, \ldots ,f^{r}$
and select from this  basis the  subbasis  consisting of  third-order iterated Lie brackets.
If we put in order   the generators  $f^{1} \prec f^{2} \prec \cdots \prec f^{r}$;
then one may think that
this   subbasis  consists of all Lie brackets
$[f^{s}[f^{i},f^{j}]]$ for which $i \leq s$ and $i < j$.

Each Lie bracket of this latter form   is involved into  Jacobi identity
$[f^{s}[f^{i},f^{j}]]-[f^{j}[f^{i},f^{s}]]+[f^{i}[f^{j},f^{s}]]=0$.
This identity  degenerates into a trivial relation of antisymmetry
(i.e. does not provide a true relation) if $i=s$ or $j=s$. Otherwise the first and
the second summands  are  elements  of the Ph.Hall basis.  The third
summand, written the way it is, if $j < s$,  or written as $-[f^{i}[f^{s},f^{j}]]$,
if $s < j$,  is a linear combination of the first and the second summands.

Let us consider a maximal stratum of $\cal S$ (consisting
of rank-(r-1) matrices) and assume, not
loosing generality that,  this stratum consists of
the matrices $C$ whose left upper $(r-1) \times (r-1)$-minor
is non vanishing: $\det A_{r-1} \neq 0$. Then
\[ C=
\left(\begin{array}{cc}
A_{r-1} & a \\
B & b
\end{array}\right)
\]
and the subcolumn $b$ is determined by the other elements of the matrix
as $b=BA_{r-1}^{-1}a$.

The elements of this subcolumn
correspond to the iterated Lie brackets  $[f^{r}[f^{i},f^{j}]]$ with
all $(i,j) \in \Gamma \subset n^{-1}(\{r, \ldots ,\frac{r(r-1)}{2}\})$, and all these  brackets
 are the elements of  the    Ph.Hall basis.
Therefore they are linearly independent
as elements of free Lie algebra, but we have to prove that
for a generic germ of distribution the linear span of their values at $q$  is transversal to  $\cD^2_{q}$.

To this end  let us choose an adopted coordinate system $(x,y):{\cal O} \rightarrow  R^n$ such that :

i) $(x,y)(q)=0_{R^n}$,

ii) $(x,y)_*|_{q}f_i(q) \
(i=1, \ldots , r)$ coincide with the first $r$ elements
$e_i , i=1, \ldots ,r,$ of the standard basis of $ R^n$;

iii) $(x,y)_*|_{q}[f_i,f_j](q)
(1 \leq i < j \leq  r)$ coincide with the next $r(r-1)/2$
elements $e_{ij} \ (i <j)$ of this standard basis.
(Obviously
$(x,y)_*|_{q}\cD^2_{q}=
 R^r \oplus  R^{r(r-1)/2}$.)

iv)  $\Delta^3_{q}= \span\{\cD^2_{q},
[f^r,[f^i,f^j]](q), | (ij) \in \Gamma \}$ is mapped isomorphically by  the differential      $x_*|_{q}$
onto a subspace of $ R^n$, while $y_*|_{q}\Delta^3_{q}=0$.

 Evidently if $\Delta^3_{q}=T_{q}M$,  this would mean that
$\{[f^r,[f^i,f^j]](q)| \   (ij) \in \Gamma \}$
span a subspace  transversal
to $\cD^2_{q}$ in $T_q M$ and then we are done.
Otherwise
choose any  minimal
subset $\bar{\Gamma} \subseteq \Gamma$ such that
$\Delta^3_{q}$ is spanned by
 $\{[f^r,[f^i,f^j]](q), | (ij) \in \bar{\Gamma} \}$
and  $\cD^2_{q}$.
Let
$y=(y_1, \ldots , y_s); \ s =\codim \Delta^3_{q}$.
Obviously
$$s=\codim \cD^2_{q}- \sharp \bar{\Gamma}
\leq (r-1)(r-2)/2- \sharp \bar{\Gamma}=\sharp(\Gamma \setminus \bar{\Gamma}).$$
Let us choose an arbitrary injection
$\sigma : \{1, \ldots , s\} \rightarrow  \Gamma \setminus \bar{\Gamma}$
and define  vector field
$$\tilde{f}^r=f^r-\sum_{\alpha =1}^s
\varepsilon_\alpha x_{\sigma(\alpha)}\partial / \partial y_\alpha,  \   (\varepsilon_\alpha \in  R,  \
\alpha=1, \ldots , s).$$
Direct computation gives us
$[\tilde{f}^r,[f^i,f^j]]=[f^r,[f^i,f^j]]$, if
$(ij) \not\in \sigma(\{1, \ldots , s\})$, in particular if $(ij) \in \bar{\Gamma}$.
Hence the values (mod$\cD^2_{q}$) at $q$ of these iterated Lie brackets   span $\Delta^3_{q}/\cD^2_{q}$.
For the pairs $(ij) \in \sigma(\{1, \ldots , s\})$ we obtain
 $[\tilde{f}^r,[f^i,f^j]]=[f^r,[f^i,f^j]]+
\varepsilon_{\sigma^{-1}(ij)} \partial /\partial y_{\sigma^{-1}(ij)}$. As we know
$[f^r,[f^i,f^j]](q) \in  \Delta^3_{q}$  and the span
of $\partial /\partial y_\alpha|_q, \ \alpha=1, \ldots ,s$
is transversal (in fact complement) to $\Delta^3_{q}$.
Therefore we constructed a distribution
$\tilde{\cD}$ generated by
$f^1, \ldots , f^{r-1},\tilde{f}^r$  which meets
 the transversality property
and is arbitrarily close to $\cD$ in $C^\infty$ topology
provided
$|\varepsilon_\alpha| \ \alpha=1, \ldots ,s$
are sufficiently small.     $\Box$







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