A classical morse theory question

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SUMMARY

The discussion centers on the relationship between the unstable manifold W^u(x) and the tangent space T_xW^u(x) in the context of Morse theory. It is established that for a non-degenerate critical point x of a Morse function f: M -> R, the tangent space at x to W^u(x) equals the eigenspace corresponding to the negative eigenvalues of the Hessian H^2f(x). The Morse lemma facilitates a quadratic decomposition of f, allowing M to be expressed as a union of unstable manifolds W^u(x). Understanding T_xW^u(x) as the tangent plane rather than points that "repel" from x is emphasized as a clearer approach.

PREREQUISITES
  • Morse theory fundamentals
  • Understanding of manifolds
  • Knowledge of Hessians in differential calculus
  • Familiarity with eigenvalues and eigenvectors
NEXT STEPS
  • Study the Morse lemma in detail
  • Learn about the properties of unstable manifolds in dynamical systems
  • Explore the computation of Hessians for various functions
  • Investigate the implications of eigenvalues in the context of critical points
USEFUL FOR

Mathematicians, particularly those specializing in differential topology, dynamical systems, and Morse theory, will benefit from this discussion.

HMY
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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?
 
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HMY said:
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?

It would be better to think that we find the equation of the tangent plane rather than points which 'repel' from x. We'd follow exactly the same steps.
 

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