# Tag Info

• 8,493
Accepted

### Understanding Morse homology with the example of a circle

Let's solve the issues one by one. Let's also use the case of a deformed sphere with the height function as the Morse function, which is similar to the circle but has more structure so that we ...
• 34.1k

### Geometric Intuition of Eigenvalues of Hessian Matrix

The comments to your question cover it pretty well, but the key piece that I think you might be missing is that (assuming appropriate conditions on the derivatives) the Hessian matrix $H$ is symmetric....
• 53.5k
Accepted

### Extensions of vector fields and the Hessian

An extension of $v \in T_pM$ to a vector field $V$ is just any $V \in \Gamma(TM)$ such that $V_p = v$. There are many extensions of a given vector, which is why we must show that the result is well-...
• 35.2k
Accepted

Accepted

### A closed manifold has closed geodesics of at most countably many lengths

It's been a while (well, 20 years) so I won't be able to fill in all the details, but the following should allow you to figure this out. Maybe someone who is still working in the area can provide an ...
• 22.3k

### Is any vector field without periodic orbits a gradient field?

This is false. Any function on a closed manifold must have critical points (for example corresponding to the maximum or the minimum), hence the corresponding gradient vector field must have equilibria....
• 9,983
Accepted

### Intuitive explanation of singular homology

Singular homology groups are the most fundamental algebraic invariant of topological spaces. A traditional but terrible metaphor claims that the singular homology groups measure the presence of holes ...
• 52.1k

### "Simple to state, but difficult to solve" problems which require analyzing topology of simplicial complexes?

A lot of work on the evasiveness conjecture uses topology of simplicial complexes. Here is a bit of an overview of the evasiveness conjecture: Suppose we have a graph on $n$ vertices and we want to ...
• 61.5k

### Morse functions with minimal number critical points

Your first question: no. A Morse function on $RP^1 \cong S^1$ has at least two critical points (a maximum and minimum). More generally, the Morse inequalities tell us that the number of critical ...
• 3,049

### Can a real-valued function on the sphere have exactly 2 critical points which are not antipodal?

I think that $f_{A,B}(C) = \frac{d(A,C)^2}{d(A,C)^2+d(B,C)^2}$ should do it. This may help with the intuition: Consider the Earth as $S^{d-1}$ with $d=3$. Let $f$ be a function that, given a point on ...
• 12.2k
The matrix $M=\{H_{i,j}(0)\}_{r\le i,j\le n}$ being symmetric and nondegenerate means it defines an indefinite inner product on $\mathbb{R}^{n-r+1}$. Hence there exists some $y\in \mathbb{R}^{n-r+1}$ ...