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Functional Analysis and Its Applications, Vol. 32, No. ?, 1998 Towards the Legendrian Sturm Theory of Space Curves \Lambda
 

Summary: Functional Analysis and Its Applications, Vol. 32, No. ?, 1998
Towards the Legendrian Sturm Theory of Space Curves \Lambda
V. I. Arnold UDC 517.9
x x x1. Introduction
Sturm theory extends the Morse inequality (minorating the number of critical points
of functions on a circle) to the higher derivatives. The Legendrian Morse theory (created
by Yu. V. Chekanov in 1986) provides the Morse inequality for the multivalued func­
tions (corresponding to the unknoted Legendrian submanifolds of the space of 1­jets of
functions). It is a generalization of the Lagrangian intersection theory due to Conley,
Zehnder, Chaperon, Floer, Sikorav, Laudenbach, Hofer, Gromov, and others. Below an
attempt to extend the Legendrian Morse theory to higher derivatives is presented. It
extends the Legendrian Morse theory in the same sense in which the ordinary Sturm
theory extends Morse theory. Sturm theory implies the existence of at least four flatten­
ing points on every curve in projective 3­space having a convex projection. The present
paper results imply, for instance, the preservation of at least four flattening points under
(not necessary small) deformations of the standard curve of Sturm theory (x = cos t ,
y = sin t , z = cos 2t), provided that the curve has no inflection (zero curvature) points
at any moment and that the Legendrian knot type remains unchanged under the de­
formation. The Legendrian knot is a singular surface in the five­dimensional manifold
of the contact elements of the space, associated to the curve. In terms of the front of

  

Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute

 

Collections: Mathematics