Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Compute $\int_M \omega$

Let $M=\{(x,y,z): z=x^2+y^2, z<1\}$ be a smooth 2-manifold in $\Bbb{R}^3$. Let $\omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in \Omega^2(\Bbb{R}^3)$. Compute $$\int_M \omega.$$ I parametrised ...
5
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0answers
55 views

Manifold, exist smooth nonnegative function with regular value at 0?

If $X$ is any manifold with boundary, then does there exist a smooth nonnegative function $f$ on $X$, with a regular value at $0$, such that $\partial X = f^{-1}(0)$?
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1answer
27 views

Quotient group and kernel of canonical projection

Imagine we have a group $G$ acting properly and freely (as a group action $\Phi: G \times M \rightarrow M$) on a manifold $M$, then $M/G$ is a manifold and there is a smooth submersion $\pi: M ...
4
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0answers
112 views

Index of zero of a gradient vector field at a critical point

Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$. How do you show that a critical point ...
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4answers
118 views

Exercise $2.1.1$ - Differential Topology by Guillemin and Pollack

If $U \subset \mathbb{R}^k$ and $V \subset H^k$ are neighborhoods of $0$, prove that there exists no diffeomorphism of V with U. Here, $H^k$ is simply the upper half-space. I tried to solve this ...
2
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2answers
43 views

Compute the tangent space at the unit matrix

Compute the tangent space $T_pM$ of the unit matrix $p=I$ when $$(i)\,M=SO(n)\\ (ii)\,M=GL(n)\\ (iii)\,M=SL(n).$$ My attempt: I think I have computed the tangent space in the case that $M=SL(n)$. ...
8
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1answer
63 views

Number of smooth structures on $\mathbb{R}$ (not up to diffeomorphism)

On page 53 of Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1, Exercise 2-4 asks How many distinct $C^\infty$ structures are there on $\mathbb{R}$? (There is only one up ...
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2answers
162 views

What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration?

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? Many thanks in advance.
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0answers
30 views

What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? [duplicate]

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? I not a topologist by trade and I find the concept kind of hard to understand... Thanks ...
2
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2answers
47 views

Adjoint representation

I was just wondering why the adjoint representation of the Lie group $Ad$ and Lie algebra $ad$ are called representation. Maybe this word is derived from abstract algebra somehow, but I don't ...
3
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2answers
73 views

$x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=0 $ $\rightarrow$ $f\equiv c$

for $f:\Bbb{R^2}\to\Bbb{R}, f\in C^1 $ and $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=0 $ for every $(x,y)\in \Bbb{R^2}$ then $f\equiv c\in\Bbb{R}$ I was thinking it's similar ...
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2answers
41 views

Is Fermat's theorem about local extrema true for smooth manifolds?

Let $M$ be a smooth manifold and $f\colon M \rightarrow \mathbb{R}$ a smooth function. If $p\in M$ is a local extremum of $f$, does $p$ have to be a critical point?
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1answer
20 views

Equivalence class of differential structures.

Given $X$ a manifold, two smooth atlas are equivalent if the union of them forms an smooth atlas. I am trying to prove that this is a equivalence relation, but I am having troubles proving that ...
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1answer
31 views

A question about an explanation of the image of an immersion f:X$\to$Y need not to be a submanifold of Y

The explanation is: From the Local Immersion Theorem,it is evident that f maps any sufficiently small NBHD W of an arbitrary point x diffeomorphically onto its image f(W) in Y. So every point in the ...
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1answer
26 views

Exercise 1.8.6 - Differential topology (Guillemin and Pollack)

Here the problem : 1.8.6 - A vector field $\vec{v}$ on a manifold $X$ in $\mathbb{R}^N$ is a smooth map $\vec{v}:X \to \mathbb{R}^N$ such that $\vec{v}(x)$ is always tangent to X at x. Verify that ...
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31 views

Regarding the construction of the tensor bundle

Recall the construction of the tangent bundle: we write $$TM = \bigsqcup_{p \in M}T_p M$$ and define it as the prevector bundle with local trivializations $[\gamma] \mapsto (\gamma(0), (x\gamma)'(0))$ ...
3
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1answer
53 views

Am I right about this definition of submanifold?

Consider the following definition of submanifold: 1.5. $\ \bf Definition.\ $ A subset $M\subset\mathbf R^{n}$ is called a $\underline{\text{differentiable submanifold}}$ of $\mathbf R^n$ of ...
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42 views

Differential forms defined by integration

Let $\omega_1,\omega_2$ be two n-forms on a $n$-dimensional manifold $M$. Now, imagine we have for every open $N \subset M$ that $$\int_{N}\omega_1 = \int_N \omega_2.$$ Can anybody show me how to ...
5
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1answer
139 views

This theory proof about instability of a point of equilibrium is not understandable for me, any help?

-This theory is irritating me, because I don't understand it's logic. Theorem: If in some neighboorhood $\mathbb O (0)$, exists a continuous, differentiable function $V(X), V(0)=0,$ such that the ...
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0answers
107 views

closed form is exact in euclidean space

Question is to show that $d(f)=0$ for a $0$ form on $\mathbb{R}^n$ then $f$ is a constant function. See that $$0=df=\sum_i\frac{\partial f}{\partial x_i}dx_i$$ implies that $\frac{\partial ...
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0answers
35 views

Reference for theorems in Hirsch

In Hirsch's differential topology, we find the following theorems on page 31: 4.1 Theorem: Let $M$ be a $C^r$ $\partial$-manifold and $N$ a $C^r$ manifold, $r \geq 1$. Let $f : M \to N$ be a $C^r$ ...
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2answers
69 views

Given two points in a manifold, can i find compact path-connected set that contains both

Suppose we are given two points in path-connected smooth manifold. My hypothesis is that we can find path-connected compact set that contains both. I have no idea how to prove it, in fact I don't know ...
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1answer
42 views

Invariance of form under flow

Take a two-dimensional symplectic submanifold $M$ in $\mathbb{R}^3$. Now, I want to show that the symplectic form $\omega$ is invariant under the Hamiltonian flow $\phi^{t}: M \rightarrow M.$ Is it ...
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1answer
40 views

$S^n$ is not a retract of the disk $D^{n+1}$ and Brouwer's Fixed Point Theorem.

I was trying to understand Hirsch's proof of this fact: "There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by ...
3
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1answer
65 views

Degrees of maps $RP^{2n+1}\rightarrow RP^{2n+1}$

What degrees are possible for maps $\mathbb R \mathbb P^{2n+1} \rightarrow \mathbb R \mathbb P^{2n+1}$? I'm asking about the odd dimensions because we cannot define degree (in a way that would make ...
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0answers
39 views

Mistake in book on symplectic topology?

I just read the proof of the non-squeezing theorem in "Introduction to symplectic topology" by Mc Duff and Salamon. The thing that is strange is that they say: Let $\Psi$ be the linear transform ...
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1answer
52 views

Two definitions of conormal bundle

Suppose $X$ is a smooth manifold and $f: Z \to X$ is an immersion with transverse self-intersection. I've seen (e.g. here) the conormal bundle to $Z$ in $T^* X$ defined as Definition A: $\quad L_Z := ...
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0answers
27 views

Zeroes of differential form embedded

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ ...
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33 views

Question on symplectic geometry

I am currently reading this paper on symplectic geometry. It deals with the question how the stability properties of a sequence of (periodic) points or fixed points can be related to the second ...
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24 views

Leray-Hirsch theorem for cohomology modulo torsion

Suppose $X,Y$ are smooth manifolds, $H^*(X,\mathbb{Z})$ is finitely generated. (*)Why do we have isomorphism modulo torsion: $H^n(X\times Y,\mathbb{Z})=\oplus_{p+q=n}H^p(X,\mathbb{Z})\otimes ...
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1answer
59 views

Intuitively, what is the difference between homeomorphism and diffeomorphism? Significance?

As the title suggests, intuitively, what is the difference between homeomorphism and diffeomorphism? Many thanks in advance. What is the significance of such a difference?
3
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0answers
33 views

Is there a version of Whitney Embedding's theorem for infinite dimensional?

Since we have a infinte dimensional version of Sard's Theorem and Transversality's theorem, I was thinking that this question is reasonable. Any reference would be greatly appreciated!
2
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2answers
32 views

Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$ [duplicate]

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
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0answers
32 views

isotopy equivalence of maps

In the book Encyclopaedia of Mathematics, Vol. 6, Question: I do not understand the part Does this mean $F_1\circ f_0=f_1: X\to Y$ or as subsets of $Y$, $$ \{y\in F_1(f_0(X))\}=\{y\in f_1(X)\}? ...
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1answer
42 views

Can the twisting of mobius band be represented by a U (1) bundle?

With the usual embedding of a mobius band, the strip is twisted by an angle pi, smoothly, as it goes round.I think this can be represented intrinsically, independent of the embedding, by attaching a ...
2
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1answer
35 views

Homotopy of certain maps induced homotopies

Let $\psi$ be a homeomorphism and $\gamma :[0,1] \rightarrow \mathbb{R}^2$ a path. Now assumme additionally that $(\psi \circ \gamma)(t) \neq \gamma(t)$ everywhere. Then we can look at the map ...
4
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2answers
113 views

Geometry of images of maps $f: \mathbb R \to \mathbb C$?

I am having trouble seeing what a continuous map $f: \mathbb R \to \mathbb C$ might look like. If it was linear it would look like a line but it's not clear to me what happens if it's any map. I ...
5
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0answers
61 views

Rigorous definition of an embedded connected sum.

Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well ...
3
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1answer
54 views

Proving that a polilinear operator is differentiable

A Polilinear map operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
3
votes
1answer
64 views

Is Heisenberg group Euclidean?

I'm reading an article speaking about Heisenberg group $\mathbb H^n$ and some of its properties. Now, I have some questions to ask, hoping to be clear enought. Reading the introduction I've ...
2
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1answer
88 views

Easy classical physics made mathematically rigorous!

Consider the following: We are given a symplectic manifold $M$. Now, we define a Hamilton function $H : M \rightarrow \mathbb{R}.$ Additionally, we want that $H^{-1}(x)=:M_x$ is a submanifold. We can ...
2
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1answer
83 views

Doesn't $x^3+2y^3+3z^3=0$ give a surface in $R^3$?

In my last exam on Advanced Calculus (following Spivak's Calculus on Manifolds), I couldn't solve the following question. True or false: the set $S$ in $R^3$ given by $x^3+2y^3+3z^3=0$ is a ...
5
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0answers
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Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
3
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1answer
58 views

Tangent space of quotient space

Let $\pi : M \rightarrow M/G$ be the canonical projection, where $M$ is a manifold and $M/G$ is a quotient manifold. Now, what can we say about $d \pi (p) : T_pM \rightarrow T_p(M/G)$? From my ...
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1answer
30 views

Lie group and stabilizer quotient

Let $G$ be a Lie group and $$G_x:=\{g \in G; Ad^*_g(x)=x\}$$ the stabilizer, where $Ad_g^*$ is the adjoint of the adjoint representation. Now, I was wondering why $G/G_x$ has a manifold structure. ...
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1answer
50 views

Does the Morse-Bott index of a critical point depend on the choice of metric?

By the Morse lemma there exists a coordinate chart $(x_1,...,x_n)$ in the neighbourhood of a critical point $p$ of a Morse function $f : M^n \to \mathbb{R}$ such that \begin{equation*} f(x) = f(p) - ...
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Condition (C) of Palais-Smale

In Klingenberg's Notes, he makes the following definition: $\Lambda M$ will be said to satisfy the condition (C) of Palais-Smale if: Given a sequence $\{c_m\}$ on $\Lambda M$ satisfying: ...
3
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1answer
43 views

Diffeomorphism between $\Bbb{R}^{4}$ and the cube

I'm looking for an explicit diffeomorphism between the four-dimensional euclidean space $\Bbb{R}^{4}$ and the four-dimensional open cube. I wonder whether there is a simple looking map, with simple ...
1
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1answer
54 views

Orbits form a manifold?

A prominent example are the coadjoint orbits $O_x = \{Ad_u^*(x);u \in G\}$ where $x \in \mathfrak{g}$ and $G$ a Lie group with Adjoint map $Ad.$ Could anybody give me an easy argument why $O_x$ is a ...
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0answers
41 views

Exponential map only for matrix Lie algebras?

Recently, I stumbled over some proofs in Lie algebra theory and noticed that they often use the notion of an exponential map $e^{t \zeta}$ for $\zeta \in \mathfrak{g}$ such that $e^{t \zeta} \in G$ ...