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Intersection of stable and unstable manifolds for invariant Morse function. (English) Zbl 1292.57027
Let \(M\) be a compact \(n\)-dimensional Riemannian manifold and \(\Phi\) be a Morse function on \(M\). For a point \(p\in M\), denote by \(\gamma _p(t)\) the corresponding negative gradient flow which passes through \(p\) at \(t=0\). If \(p\) is a critical point of \(\Phi\), then \(W^u(p)\) and \(W^s(p)\) denote the unstable manifold of \(p\), and the stable manifold of \(p\), respectively. The main result of the paper is contained in Theorem 1: Assume that \(M\) admits an action of a compact connected Lie group \(G\) and \(\Phi\) is a \(G\)-invariant Bott-Morse function on \(M\). Let \(p,q\) be \(G\)-fixed points and suppose the following conditions hold : (1) \(M^G \subset Cr(\Phi)\); (2) \(\lambda(p)- \lambda(q)=2\), where \(\lambda(p)\) denotes the Morse index of \(p\); (3) \(W^u(p)\) and \(W^s(q)\) intersect transversally. Then every connected component of \(\widetilde{\mathcal{M}}(p,q)\) is diffeomorphic to \(S^1\times \mathbb{R}\), where \(\widetilde{\mathcal{M}}(p,q)=W^u(p)\cap W^s(q)\).
MSC:
57R70 Critical points and critical submanifolds in differential topology
57S15 Compact Lie groups of differentiable transformations
37D15 Morse-Smale systems
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