A classical morse theory question

In summary, we have a manifold M with a Morse function f. We consider a critical point x and its unstable manifold W^u(x) under the negative gradient flow on M. The tangent space at x to W^u(x) is equal to the eigenspace of the Hessian H^2f(x). With the assumption that all critical points are non-degenerate, the Morse lemma allows for a quadratic decomposition of f. Additionally, M can be written as the union of all unstable manifolds W^u(x). The tangent space T_xW^u(x) can be thought of as the equation of the tangent plane rather than points that repel from x.
  • #1
HMY
14
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?
 
Physics news on Phys.org
  • #2
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.
 

1. What is classical Morse theory?

Classical Morse theory is a mathematical theory that studies the topology of smooth manifolds by using the critical points of a smooth function defined on the manifold.

2. How is classical Morse theory used in science?

Classical Morse theory has applications in many fields of science, including physics, engineering, and biology. It can be used to study the shape of molecules, the behavior of fluids, and the stability of dynamical systems, among other things.

3. What are the main concepts in classical Morse theory?

The main concepts in classical Morse theory include critical points, index theory, and the Morse inequalities. Critical points are points where the derivative of a smooth function is zero, and index theory assigns a number to each critical point to describe its behavior. The Morse inequalities relate the number of critical points of different indices to the topology of the manifold.

4. Are there any practical applications of classical Morse theory?

Yes, classical Morse theory has many practical applications, such as finding optimal paths for vehicles or robots, understanding the dynamics of biological systems, and identifying patterns in data. It can also be used in image processing, computer vision, and machine learning.

5. Are there any modern developments in classical Morse theory?

Yes, there have been many modern developments in classical Morse theory, including extensions to non-smooth functions, applications in symplectic geometry and mathematical physics, and connections to other areas of mathematics such as algebraic geometry and topology. There are also ongoing research efforts to generalize and refine classical Morse theory for more complex systems.

Similar threads

I Prove $$T_{p}M$$ is a vector space with the axioms
    • Differential Geometry
Replies
3
Views
1K
I Integrability of the tautological 1-form
    • Differential Geometry
Replies
1
Views
1K
I The Road to Reality - exercise on scalar product
    • Differential Geometry
Replies
2
Views
575
A Why are sheafs defined using abelian groups?
    • Differential Geometry
Replies
8
Views
2K
I Induced orientation on boundary of ##\mathbb{H}^n## in ##\mathbb{R}^n##
    • Differential Geometry
Replies
6
Views
315
B Relating basis vectors at different points in a neighborhood
    • Differential Geometry
Replies
5
Views
1K
B Reconciling basis vector operators with partial derivative operators
    • Differential Geometry
Replies
4
Views
1K
A Hamiltonian formulation of classical mechanics as symplectic manifold
    • Classical Physics
Replies
5
Views
385
I Tangent vector basis and basis of coordinate chart
    • Differential Geometry
Replies
6
Views
3K
A Question about the derivation of the tangent vector on a manifold
    • Differential Geometry
Replies
9
Views
3K

Hot Threads

Recent Insights

Back
Top