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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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##### References:
  M. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1-15. · Zbl 0482.58013  A. Banyaga and D. Hurtubise, Lectures on Morse Homology , Kluwer Texts in the Mathematical Sciences, Volume 29 (2004). · Zbl 1080.57001  M. Goresky, R. Kottwitz and MacPherson, Equivariant cohomology, Koszul duality and the localization theorem, Invent. Math. 131 (1998), no. 1, 25-83. · Zbl 0897.22009  J. F. Plante, Fixed points of Lie group actions on surfaces, Ergod. Th. and Dyn. Sys. 6 (1986), 149-161. · Zbl 0609.57020  E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), no. 4, 661-692. · Zbl 0499.53056
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