The morse-theory tag has no wiki summary.
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21 views
Need an application of Morse theory for second-order differentialle systems
I'm looking for some applications of Morse theory for the second order differentialle systems,
Someone can help me with a pdf or a book or an article which has these applications ?
Please
Thank ...
0
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1answer
32 views
Question on the demonstration of Morse theorem
We have theorem of Morse
and this is the proof
i dont understand this :
"$(c_i)$ has no cluster point since each $M^a=f^{-1}]-\infty,a]$ is compact "
Thank you.
1
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2answers
23 views
Question about index of critical points.
I don't really understand what index of a critical point is and I am trying to do a very simple example. I was wondering if someone could help me figure out what the index of the critical point ...
3
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1answer
181 views
Question about theorem 3.2 from Morse theory by Milnor
THe demonstration of the theorem 3.2 in the book Morse theory by Milnor
is given in the special case whene the manifold is the Torus ,
My question is : can i prove it in the case where the ...
2
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1answer
127 views
An other question about Theorem 3.1 from Morse theory by Milnor
In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that:
for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla ...
0
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1answer
25 views
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
3
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1answer
118 views
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 ...
2
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1answer
89 views
Question on Theorem 3.1 from Morse theory by Milnor
In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that:
for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla ...
5
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1answer
90 views
How to check whether a vector field is Morse-Smale?
Setup and notation:
Let $f:M\to \mathbb{R}$ be a Morse-function on the compact $m$-dimensional manifold $M$ and let $X$ be a gradient-like vector field for the function $f$.
Denote the unstable ...
2
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0answers
56 views
Prove Poincare duality theorem with Morse theory.
First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
1
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0answers
59 views
About the Morse theory
I am trying to study the Morse theory and would like to know the purpose of this study, why when we talk about the critical point of a manifold is mentioned Morse theory? , are the critical points of ...
2
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0answers
44 views
Proof of the Morse Lemma in dimension 1
This is the Morse lemma in dimension1 :
Let $M$ be a smooth $1$-manifold and $f: M \longrightarrow \Bbb R$ be a smooth function. Suppose $p$ is a non-degenerate critical point of $f$.
Then ...
1
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1answer
47 views
Lemma2.1 (in dimension 1)in the book of Morse theory by Milnor
i have this lemma :
Let $f$ be a $C^{\infty}$ function in a convex neighborhood $V$ of
$0$ in $\mathbb{R}$ , with $f(0)=0$ then $f(x)= x g(x)$. for suitable
$C^\infty$ function $g$ defined in ...
0
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1answer
27 views
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 ...
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1answer
43 views
0
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1answer
51 views
Non-degenerate smooth functions on a manifold
I am trying to prepare a presentation on "the use of differential geometry in the theory of critical points", but only the case where there is a single variable. (only in dimension 1),
and i ask ...
11
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1answer
182 views
Gradient-like vector fields
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 ...
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2answers
68 views
Question on Morse Theory.
I am studying Morse Theory, and I would like to know what a ‘non-degenerate smooth function’ means. Please help. Thanks!
2
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0answers
35 views
Set of Morse-Smale Functions is Dense in Euclidean Space
In much of the Morse Theory literature, it seems that we always assume $M$ is a smooth closed (or compact) manifold so that we get theorems like:
The set of Morse-Smale gradient vector fields on ...
4
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0answers
59 views
Existence of covers on manifolds with certain properties
I'm trying to prove the existence of Morse functions on differentiable manifolds, by adapting the proof found on Matsumoto's textsbook, which works for compact manifolds, to the non-compact case.
I ...
1
vote
2answers
102 views
How to obtain a Morse function on a submanifold of Euclidean space
Consider a smooth $n$-dimensional submanifold $A$ in $\mathbb{R}^{n+1} \times \mathbb{R}$ and the projection $f:\mathbb{R}^{n+1} \times \mathbb{R}\rightarrow \mathbb{R}$ onto the second factor. Is it ...
6
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1answer
94 views
Morse homology of $P^2$
I have seen and worked through the explicit computation of the Morse homology of the sphere and the torus (with signs and all). But trying it for $\mathbb{R}\mathbb{P}^2$ has lead me to dead ends. Is ...
1
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1answer
57 views
The connection of Morse function
Suppose $M$ and $N$ are two manifolds, $f$ is a Morse function on $M$, $g$ is a Morse function on $N$, can you find a new manifold $P$ as the connection of $M$ and $N$ and a Morse function $h$ on the ...
5
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2answers
270 views
Is there a function like this?
Let $A=[0,1]$ and $C=\{0\}\cup\{\frac{1}{n},\ n\in\mathbb{N}\}$.
i) Is there a function $f:A\rightarrow\mathbb{R}$ such that $f\in C^{r}(A)$, $r\geq 2$ and the set of critical "Values" of $f$ is $C$?
...
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0answers
49 views
Some examples for the Morse theory
In the Morse theory, we have some powerful theorems, such as the existence of the self-index Morse function and the Morse-Smale pair on the smooth compact manifold. But that is a general way, does any ...
