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Good night, anyone know of any reference where I can find the proof of the Stable/Unstable Manifold Theorem for a Morse-Bott function. I'm interested in the dimensions of the stable and unstable manifolds, these dimensions are intuitive but wanted to know of some reference.

Thank you very much, all references will help me.

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The paper Morse-Bott theory and equivariant cohomology by D. M. Austin and P. J. Braam (in The Floer memorial volume, Progr. Math., 133, 1995) is a good reference for Morse-Bott theory. In particular Proposition 3.2 and Theorem A.9 seem to contain what you want.

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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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As far as I remember (I don't have the book at hand right now to check), the book Differential Equations and Dynamical Systems by Lawrence Perko (Springer, 1991) has a version of the stable manifold theorem for the case that some eigenvalues of the linearization of your vector field have zero real part.

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