Legendrian Sturm theory. V.Arnold.
    Summary of the talk on 12.02.97

The Legendrian Morse theory of Chekanov (1985) is an extension of the
Lagrangian intersection theorem (conjectured itself in 1965 as an
extension of the Morse theory to the multivalued functions and proved
in 1983-4 by Conley,Zehnder,Chaperon,Floer etc).
      Chekanov extended their theory of intersections of embedded
Lagrangian submanifolds with the zero section of the cotangent
bundle space to some immersed Lagrangian manifolds- namely to those,
which one can reach from the zero section by a regular homotopy
preserving the type of the corresponding Legendrian knot (in the space
of 1-jets of functions).
   In this talk I shall show how to extend similarly the Sturm-type
theorem on the four flattening points on a curve in the projective
3-space.The extended theorem minorates the number of the swallowtails
on the fronts of the Legendrian surfaces,"unknotted" in the
contact 5-space.
    The four vertices (flattenings, swallowtails) are provided by
the following combinatorial
    Lemma. Consider a smooth curve,cutting the Moebius band into oriented
domains (the orientation being different from both sides). Then the curve
contains at least two components,intersecting the central circle of the
band an odd number of times.