Generalized billiard paths and Morse theory for manifolds with corners
Let \(M\hookrightarrow \mathbb{R}^N\) be a smooth \(n\)-manifold embedded in Euclidean space dimension \(N\). If \(p,q\in \mathbb{R}^N\), then a generalized billiard path with \(k\)-reflections connecting \(p\) to \(q\) is an ordered sequence \(P=\{\alpha_1, \dots,\alpha_k\}\) of points \(\alpha_1,\dots,\alpha_k\in M\) such that for each \(i=1,\dots,n\), one of the following is true: (1) The bisector of \(\sphericalangle \alpha_{i-1}\alpha_i \alpha_{i+1}\) is normal to \(T_{\alpha_i} M\). (2) \(\sphericalangle \alpha_{i-1} \alpha_i \alpha_{i+1}\) is a straight angle where \(\alpha_0=p\) and \(\alpha_{k+1}=q\). This definition allows the line segment \(\overline{\alpha_i \alpha_{i+1}}\) to intersect the manifold. If \(M\) happens to be a convex hypersurface, the definition reduces to the usual notion of billiard path. The author desires to estimate the number of generalized billiard paths connecting fixed points with a given number of reflections. The length of a series of line segments with \(k\) reflections may be thought of as a function on \(M^k\) and generalized billiard paths correspond to critical points of this length function. The length function is not smooth on \(M^k\), having singularities along its diagonals \(\{\alpha_i= \alpha_{i+1}\}\). But following the Fulton-MacPherson procedure [Ann. of Math. 139, 183-225 (1994; Zbl 0820.14037)] one can `blow up' \(M^k\) to obtain a compact manifold with corners \(X_k\) to which the length function extends smoothly. In this situation the author develops a version of Morse theory for manifolds with corners and uses it to study this length function. This Morse theory is a generalization of that of \textit{D. Braess} [Math. Ann. 208, 133-148 (1974; Zbl 0263.58005)], retaining both a global `gradient' flow and the intrinsic stratification of a manifold with corners. Then the author finds that the number of generalized billiard path with \(k\) reflection connecting two points in \(\mathbb{R}^N\) can be estimated in terms of the homology of the manifold \(M\). The main results are the following theorems: Theorem 21. The number of generalized billiard paths with \(k\) reflections is at least \(\sum^{kn}_{i=0} b_i(X_k)\) where \(b_i(X_k)\) denotes the \(i\)th Betti number of \(X_k\). Theorem 30. The number of generalized billiard paths connecting \(p\) to \(q\) with \(k\) reflections is at least \(\sum^{n-1}_{j=0} \sum_{i_1+ \cdots+ i_k=j} b_{i_1}(M)\dots b_{i_k}(M)\).
scientific article; zbMATH DE number 1891201
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