Summary: On topological properties of Legendre projections in
contact geometry of wave fronts
V. Arnold
January 27, 1998
There exists an interesting relation between the global topological prob­
lem of coexistence of singularities on wavefronts and the Sturm type theory
of oscillations of linear combinations of eigenfunctions of differential opera­
tors.
One can use this relation in both directions.In this paper the Sturm­
Tabachnikov theorem on the oscillations of linear combinations of eigen­
functions of differential operators on a circle is used to minorate the number
of cusps of a generic wave front close to a circle collapsing to a point.
The classical four vertices theorem for curves in the euclidean plane is
shown to be a particular case of a general fact of contact topology. This
observation suggests the higher­dimensional generalizations of the four ver­
tices theorem and also suggests some nontraditional generalizations of the
Sturm theory for the functions on spheres.
The simplest of these generalizations corresponds to the theorem on the
four umbilical points on a closed convex surface in three­dimensional eu­
clidean space.The general formulation in terms of the symplectic and contact

Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute

Collections: Mathematics