- #1

HMY

- 14

- 0

Let x be a critical point of f and assume all critical points are non-degenerate.

Let W^u(x) be th unstable manifold of x when considering the negative gradient flow on M.

Why does the tangent space at x to W^u(x) = Eig^- H^2f(x)?

Denote the Hessian by H^2f(x).

I know that since the critical points are non-degenerate the Morse lemma gives a sort

of quadratic decomposition of f. I also know the M can be written as the union over

all x of the W^u(x).

One of the problems is that I don't really understand the object T_xW^u(x). The tangent

space is a vector space. So T_xW^u(x) consist of the points in the vector space that

"repel" from x?