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.