- #1
HMY
- 13
- 0
Let M be a manifold and let f: m -> R a Morse function.
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?
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?
