# 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.

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### Normal Bundle of a Manifold

I was reading "Morse Theory" by J.Milnor and at page number 32 there is remark "It is not difficult that N is an n-dimensional manifold differentiably embedded in $\mathbb{R}^{2n}$ ( N is the total ...
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### Application of Lefschetz duality to prove Lefschetz hyperplane theorem

I'm trying to understand the proof of the Lefschetz hyperplane theorem in Milnor's book "Morse Theory", page 41 but I can't understand his use of Lefschetz duality. At this point it has been proven ...
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### Morse functions are dense in $C^{\infty}(M,\mathbb{R})$ questions.

Hi here is a proof inspired from the reference below. Feel free to get very technical with your comments so that at the end I understand it well. I am more concerned about the questions I added below,...
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### Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
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### Question on Theorem 5.1 from K.C Chang's book

I have this theorem and in the first part of the proof , I don't understand why $d_z f(\theta_1+\theta_2)=\theta_1$ ? Theorem $5.1$. Suppose that $U$ is a neighborhood of $\theta$ in a Hilbert ...
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### When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
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### Background for reading Milnor's Morse Theory book

I wish to study the book 'Morse Theory' by J. Milnor, but I am not sure whether I have the necessary prerequisites. I know basic point set topology, real analysis (limits, continuity, differentiation, ...
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### Fundamental theorem of Morse theory for $\Omega(S^n )$

Using the Fundamental theorem of Morse Theory we can prove that $\Omega(S^n)$ is homotopically equivalent to a CW complex with one cell each in dimensions $o,n-1,2(n-1), \cdots$ and so on. But how can ...
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### Finite approximation of path space.

Let $M$ be a connected Riemannian manifold, $\Omega(M)=\Omega$ the path space and $E: \Omega \to \mathbb{R}$ the energy function. We can define $\Omega^c:=E^{-1}([0,c])$ and $\Omega(t_0, \dots, t_k)$ ...
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### Path space of $S^n$

Suppose that $p,q$ are two non conjugate points on $S^n$ ($p \ne q,-p$). Then there are infinite geodesics $\gamma_0, \gamma_1, \cdots$ from $p$ to $q$. Let $\gamma_0$ denote the short great circle ...
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### Non degenerate critical points.

Let $K$ be a compact subset of the Euclidean space $\mathbb{R}^n$; let $U$ be a neighborhood of $K$ and let $f:U \rightarrow \mathbb{R}$ be a smooth function such that all critical points of $f$ in $K$...
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### How does $\operatorname{Ric} \ge 0$ guarentee the Busemann function is regular in the splitting theorem?

Cheeger-Gromoll's famous splitting theorem says If $(M,g)$ contains a line and $\operatorname{Ric} \ge 0$. Then $(M,g)$ is isometric to a product. I want to know how does $\operatorname{Ric} \ge 0$...
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### Question on Morse lemma

I have this: (Page 421, heading Asymptotically quadratic functionals) Remark 2.2. (a) If $N$ is any neighbourhood of $x_0$, then the excision property of homology theory implies C_k(f,x_0) ...
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### Retraction by deformation

in Theorem 3.1 in the book Morse theory by Milnor , in the end of the proof they say that : $r_t$ is a deformation retract ? how to prove this ? please thank you
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### Lemme 2.4 in Morse theory by Milnor

This is lemma 2.4 from "Morse theory" by Milnor ,with the prove I have some questions about this prove : 1) why $\displaystyle\frac{dc}{dt}(f)=\lim_{h\rightarrow 0} \frac{fc(t+h)-fc(t)}{h}$ and ...
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### Lemma of Morse in dimension 1

I want to write the Morse lemma which is in dimension $n$ : Let $p$ be a non-degenerate critical point for $f$. Then there is a local coordinate system $(y^1,...,y^n)$ in a neighborhood $U$ ...
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### Index of Morse function

What is the definition of the index of a Morse function in dimension one?
Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function. My goal is to better understand gradient-like vector fields for $f$. Question: Do any two ...