Generalized billiard paths and Morse theory for manifolds with corners

[摘要] A billiard path on a manifold M embedded in Euclidean space is a series of line segments connecting reflection points on M. In a generalized billiard path we also allow the path to pass through M. The two segments at a ;;reflection;; point either form a straight angle, or an angle whose bisector is normal to M. Our goal is to estimate the number of generalized billiard paths connecting fixed points with a given number of reflections. We begin by broadening our point of view and allowing line segments that connect any sequence of points on M. Since a path is determined by its ;;reflection;; points, the length of a path with k reflections may be thought of as a function on M k. Generalized billiard paths correspond to critical points of this length function. The length function is not smooth on M k, having singularities along some of its diagonals. Following the procedure of Fulton and MacPherson we may blow up Mk to obtain a compact manifold with corners to which the length function extends smoothly. We then develop a version of Morse theory for manifolds with corners and use it to study the length function. There are already versions of Morse theory that may be used in this case, but ours is a generalization of the work of Braess, retaining both a global ;;gradient;; flow and the intrinsic stratification of a manifold with corners. We find that the number of generalized billiard paths with k reflections connecting two points in RN can be estimated in terms of the homology of the Fulton-MacPherson blow up. Proceeding further, we find lower bounds for the number of such paths directly in terms of the homology of M. In part, we show the number of these paths is at least j=0n-1 i1+˙˙˙+ik=jb i1M&ldots; bikM .

[发布机构] Rice University