# Questions tagged [morse-theory]

For questions about Morse theory, which is a branch of differential topology to analyze topology of manifolds by studying differentiable functions on them.

304 questions
Filter by
Sorted by
Tagged with
172 views

### (Negative) Gradient and Orientability of its flow.

Before asking my question, I put the necessary definitions and some context. If you are used with Morse Theory, you can skip the text within [[[...]]]. [[[Let me first define what I mean by gradient ...
106 views

### Computing derivative in second tangent bundle

Suppose $M$ is an $m$-dimensional manifold and $\gamma \colon \mathbb{R} \rightarrow M$ is a path with $\gamma(0) = p_0$ and $\gamma'(0) = v_0 \in T_{p_0}M$. Let $g \colon \mathbb{R} \rightarrow TM$ ...
113 views

187 views

### Morse theory: the loop space of $S^n$

I am currently learning Morse theory through Milnor's book on the subject. I am trying to understand the result given in p.96 that tells us the cell decomposition of the loop space of $S^n$: ...
87 views

### Morse theory problem II

I keep reading the book of Milnor, Morse theory and i have a problem. It exactly this one. At the very end he says that is clearly that $\varphi_{b-a}$ takes $M^a$ diffeo to $M^b$ and for me is not ...
231 views

### Extensions of vector fields and the Hessian

Im reading book of Milnor, Morse theory and at the very beginning, he defines the hessian as in the paragraph above. My question is, what is the precise definition of extension to a vector fiels of ...
13 views

### How many height function on a given compact oriantable manifold there exist up to $SO(n)$ rotations?

How many height function on a given compact oriantable manifold there exist up to $SO(n)$ rotations? It seems to me that there is only one height function, but I have no Idea how to show it.
62 views

### Energy functional is locally constant on moduli space

Let $\alpha \in \Omega^1(M)$ be a closed $1$-form on a closed Riemannian manifold $(M,g)$. Denote by $X$ the corresponding dual vector field. Consider a flow line $$\gamma \colon \mathbb{R} \to M$$ ...
148 views

65 views

### Contracting paths of framed functions

Let $M$ be a smooth manifold. Morse functions on $M$ are smooth functions $M \to \mathbb{R}$ with only very nice singularities. Fact: The space of Morse functions on $M$ is not, in general, ...
109 views

### Attaching cells in the Morse Theory

Every critical point of a Morse function tells us to glue a cell of a dimension equal to the index of the critical point. Is there any information about how to glue this cell? For instance, does the ...
64 views

### Is this a trivial case of the Morse lemma?

Say we have a $C^{\infty}$ function $f: \mathbb{R} \to \mathbb{R}$ and a point $x_0 \in \mathbb{R}$ such that $f'(x_0) = 0$ but $f''(x_0) \not= 0$. Then does the Morse lemma guarantee us that there ...
72 views

42 views

### Difference between 'similar looking' Morse functions

Let's say $f$ and $g$ are two Morse functions on same manifold $M$, and have same domain, same range, same critical points, same critical values at those points and same index at those critical points....
Let $f$ be a Morse function on closed smooth manifold $M$, (one can also suppose that this is a hyperbolic manifold) G is a group which acts effectively and smooth on $M$, f and G (hyperbolic metric) ...
I am reading Milnor's book Morse Theory on p.67 he defines tangent space. "By the tangent space of $\Omega$ (which is the path space) at a path $\omega$ will be meant the vector space consisting of ...