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In differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold.

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Stable/Unstable Manifold Theorem for a Morse-Bott function

The most general sort of theorem along those lines can be found in the book Invariant Manifolds by Hirsch, Pugh, and Shub, Springer Lecture Notes in Mathematics Volume 583.
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