# Tagged Questions

**4**

votes

**1**answer

159 views

### cell decomposition of real homogeneous hypersurfaces

Let $f(x_1,\ldots,x_n)\in\mathbf{R}[x_1,\ldots,x_n]$ be a homogeneous polynomial
and consider the real hypersurface $H=\{(x_1,\ldots,x_n)\in\mathbf{R}^n:f(x_1,\ldots,x_n)=1\}$. Assume to simplify ...

**7**

votes

**0**answers

221 views

### diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...

**6**

votes

**3**answers

200 views

### Is the set of focal points of a submanifold on a normal geodesic discrete?

Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential ...

**3**

votes

**1**answer

313 views

### Is the space of gradient-like vector fields contractible?

Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function.
Question: Is the space $GVect(M,f)$ of all gradient-like vector-fields for $f$ contractible ...

**6**

votes

**1**answer

407 views

### When is the determinant a Morse function?

This might be ridiculously obvious, but...
For each $n \in \mathbb{N}$, let $M_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dimensional determinant ...

**4**

votes

**1**answer

208 views

### Morse lemma with least amount of regularity.

I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...

**3**

votes

**1**answer

293 views

### Special Morse function on a Riemann surface

Let $f: S \to \Bbb R$ be a Morse function on a Riemann surface. Let $x_0$ be a saddle point of $f$. Since $x_0$ is a critical point of $f$, it makes sense to talk about the bilinear forms ...

**1**

vote

**4**answers

359 views

### Perturbation of Morse function

The following question is probably classic in Morse theory, so a reference to an existing result should be sufficient. I don't know much about Morse theory and I am dealing with the following ...

**30**

votes

**3**answers

1k views

### Operations via Morse Theory

I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology ...

**5**

votes

**2**answers

425 views

### Reference request: embedded Morse theory

For references of embedded Morse theory, or so-called relative morse theory of a pair, I have found R.W. Sharpe's "Total Absolute Curvature and Embedded Morse numbers" totally not helpful. For further ...

**11**

votes

**5**answers

1k views

### Yang-Mills and Chern-Simons functionals as Morse functions

Can the Yang-Mills or Chern-Simons action functionals be considered as [possibly perfect] Morse functions? I assume we would be in an equivariant scenario due to considering the configuration spaces ...

**6**

votes

**2**answers

517 views

### Morse Function on 3-Torus from Heegaard Splitting

It is easy to visualize some Morse functions on surfaces (such as the torus) via the height function, but this seemingly doesn't work for 3-manifolds. I am looking for an explicit one on the 3-torus ...

**7**

votes

**0**answers

533 views

### Good introduction to Morse-Novikov theory?

Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued ...

**14**

votes

**1**answer

845 views

### ``Nice'' metrics for a Morse gradient field: counterexample request

Ralph Cohen (professor at Stanford) is teaching a class on algebraic topology and moduli spaces this quarter, beginning by reviewing his perspective of Morse theory. He defined "nice" metrics, proved ...

**4**

votes

**1**answer

470 views

### Morse theory and adiabatic limits

Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and ...

**4**

votes

**2**answers

459 views

### Gluing in Morse homology for Hilbert manifolds

Is there any reference for gluing in the context of Morse homology on Hilbert manifolds?
Gluing is pretty standard in Morse homology for finite-dimensional manifolds. Unfortunately, in the ...

**2**

votes

**0**answers

289 views

### Is the Hessian almost everywhere nondegenerate?

Let $M$ be a complete Riemannian manifold. For a fixed point $p$ in $M$, the Riemannian distance to $p$ is denoted by $d_p$. Fix a strongly convex geodesic ball $B(o,R)$ in $M$ and some disjoint ...

**0**

votes

**3**answers

749 views

### Index of a Morse function via the Hessian tensor

For a smooth function $f:M\to \mathbb{R}$ one usually defines the degeneracy and index of a critical point $p\in M$ in terms of the eigenvalues of the Hessian matrix $(\partial^2 f/\partial ...

**3**

votes

**4**answers

668 views

### literature on geometrical viewpoint on calculus of variations for physics

What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model?
...

**6**

votes

**2**answers

1k views

### existence of Morse functions satisfying the Palais-Smale condition

Given an arbitrary Hilbert manifold, can on find a complete Riemannian metric and a Morse function satisfying the Palais-Smale condition?

**5**

votes

**1**answer

759 views

### Morse-Bott homology for infinite-dimensional manifolds

Is there any work on Morse-Bott homology for infinite-dimensional manifolds (e.g. Hilbert manifolds). I am particularly interested in the case where we have a locally trivial fiber bundle and the ...

**9**

votes

**6**answers

2k views

### CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature.
Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...

**6**

votes

**2**answers

1k views

### Critical points on a fiber bundle

Consider a (smooth) bundle Eā_B_, and a (smooth) function f: E ā R on the total space. Then it makes sense to talk about the derivatives of f along the fibers. Let C be the subspace of E consisting ...