On three-periodic trajectories of multi-dimensional dual billiards

We consider the dual billiard map with respect to a smooth strictly convex closed hypersurface in linear 2m-dimensional symplectic space and prove that it has at least 2m distinct 3-periodic orbits.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-33.abs.htm

On skew loops, skew branes and quadratic hypersurfaces

A skew brane is an immersed codimension 2 submanifold in affine space, free from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew brane cannot lie on a quadratic hypersurface. We also prove that there are no skew loops on embedded ruled developable discs in 3-space. The paper extends recent work by M. Ghomi and B. Solomon.Comment: 13 pages, 2 figure

Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem

We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law "angle of incidence equals angle of reflection" are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. This is a magnetic analog of Hilbert's fourth problem asking to describe the Finsler metrics whose geodesics are straight lines.Comment: 27 pages, 6 figure

Converse Sturm-Hurwitz-Kellogg theorem and related results

The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preserving diffeomorphism of the circle that takes this function to a function, orthogonal to the Chebyshev system. We also prove that if a function on the real projective line has at least four sign changes then there exists an orientation preserving diffeomorphism of the projective line that takes this function to the Schwarzian derivative of some function. These results extend the converse four vertex theorem of H. Gluck and B. Dahlberg: a function on a circle with at least two local maxima and two local minima is the curvature of a closed plane curve