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Topological problems of the theory of asymptotic curves
 

Summary: Topological problems of the theory of
asymptotic curves
V.I. Arnold 1
to S.P. Novikov admiringly
I. Introduction.
An asymptotic curve on a surface in the euclidean or projective 3­space is an integral
curve of the field of asymptotic directions (directions along which the second quadratic form
vanishes). It is proved below that (typical) asymptotic curves on hyperbolic surfaces are
exactly the (typical) space curves having no flattening points. These curves might also be
defined as the smooth curves whose dual curves are also smooth. I call them rotating curves.
A rotating curve may have inflection points (where the curve curvature vanishes).
The rotating curves in the three­dimensional projective space have remarkable topological
properties. These properties imply, for instance, that an asymptotic curve of a surface, which
is a graph of a smooth function, can't be a star­like curve (this result find by D. Panov was
the starting point of the present work). In spite of this, there exist closed asymptotic curves
on some hyperbolic surfaces which are graphs of functions of two variables. Some examples
are presented below (to construct them is an unexpectedly complicated task). I do not know
what is the minimal number of the inflections points on such a curve. (Using the method of
the present paper, F.Aicardi has recently constructed curves with only two inflections).
The dynamical systems, defined by the asymptotic direction fields on the (doubled) hy­

  

Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute

 

Collections: Mathematics