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.
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.
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.