Tagged Questions

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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1answer
152 views

smoothness of hopf fibration projection with respect to standard differentiable structure on unit sphere

We know that steographic projection defines a differentiable structure on $S^n$ by sending points on $S^n$ to hyperplane $\{x^{n+1}\}=0$. In fact, stereographic projection $\sigma_P: S^n- P\to R^n $ ...
4
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2answers
263 views

How does Thurston's geometrisation conjecture imply Poincaré's conjecture?

I ran into the geometrisation conjecture a few days ago, and I started wondering how to prove Poincaré's conjecture. Let $M$ be a compact, simply connected, $3$-manifold. Clearly it is irreducible ...
11
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1answer
160 views

Orientation on finite dimensional vector spaces over finite fields.

For finite-dimensional $\mathbb R$-vector spaces, we define an orientation to be an equivalence class of ordered bases, where $B_1 \sim B_2$ iff the change of basis matrix $A$ taking $B_{1}$ to ...
0
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1answer
77 views

Tangent spaces of compact spaces

In a recent discussion of tangent spaces, it was noted that tangent spaces to a manifold are not compact because by definition they are vector spaces. I was curious as to whether tangent spaces to ...
4
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1answer
112 views

orbits are open in Manifold ? group action on manifold.

I need to show: for a differentiable manifold $M$, and $Aut(M)$ acts on $M$, orbit of a point $a\in M$ is open in $M$, please help.
11
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1answer
391 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
455 views

Help understanding manifolds and topological spaces

I'm used to think about linear algebra with matrices and vectors, I don't have particular problems with geometry either, I'm having hard times understanding what is the meaning of a manifold and a ...
3
votes
1answer
272 views

Level Sets are Regular Submanifolds

In Section $9$ of Tu's Introduction to Manifolds, we're asked to find all values $c\in\Bbb R$ for which the level set $f^{-1}(c)$ is a regular submanifold when $$f(x,y)=x^3-6xy+y^2.$$By taking each of ...
4
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1answer
105 views

Proving that the smooth automorphism group of a manifold $M$ acts transitively on $M$

Let $M$ be a differentiable manifold of dimension $n$, let $p,q\in M$ any two points. We need to show there exists an automorphism $f\in \mathrm{Diff}(M)$ with the property that $f(p)=q$. Could ...
21
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1answer
385 views

How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
4
votes
1answer
127 views

Write down the equation of the tangent plane and compute the Taylor series of the function

Set $f(x,y,z) = x + y + z + x^2 + y^2 + z^2$. Consider the surface $$S = \{f(x,y,z) = 0\} \subset \mathbb{R}^3$$ near the origin $o = (0,0,0) \in S$. Write down the equation of the ...
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0answers
64 views

morse function and immersion

I have a question from Pollack and Guillemin's book: let $X$ be a smooth manifold. $f:X\to \mathbb R^n$ an immersion, $f=(f_1,....f_n)$. show that for almost all $a_1,...,a_n$, the function ...
3
votes
0answers
104 views

Extension of Brouwer's degree to continuous functions.

I am studying the first chapter of this book: Topological Degree Theory and Applications At page 13 of this document, Definition 1.2.5, it is essentially said that to define the degree of a ...
1
vote
1answer
131 views

A question about orthogonal projections of Smooth Embeddings of the circle.

Question: Let $f$ be a smooth embedding of $S^1\rightarrow \mathbb{R}^3$. Given an element $v\in S^2$ we have the orthogonal projection $\pi_v:\mathbb{R}^3\rightarrow P_v$ to the plane $P_v$ = the ...
5
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0answers
232 views

Partial derivative of a composite function $\mathbb{R}^n \to \mathbb{R}^n$

I am trying to understand a proof but I am stuck on this technical bit: Apart from the small typo highlighted, I don't really see how to get the big formula for the partial derivative of $v_i$ ...
1
vote
1answer
91 views

Brouwer's degree: equivalent definitions

I am reading "Topological degree theory and applications" by O'Regan, Cho and Chen. I am stuck on the start: Consider $\Omega\subset \mathbb{R}^n$ open and bounded and let $f\in C^1(\bar \Omega)$, ...
2
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0answers
39 views

Topological inequivalent manifolds obtaining by removing a surface from a manifold

Are there any general techniques for classifying the inequivalent topologies that can be obtained by removing a 2-surface S from a 4-manifold M? I am particularly interest in the case where both M and ...
0
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1answer
52 views

Restricting the domain of an integral on a manifold

I would like to prove the following: Guess. Suppose M is an orientable smooth manifold without boundary and W is an open set in M. If $\omega$ is a smooth n-form on M such that $\operatorname{supp} ...
2
votes
2answers
121 views

Smooth maps between Euclidean spaces

There is a question that has bothered me for quite a long time. When we define smoothness of a function(or a map from one space to another), we define it as "has continuous partial derivatives of all ...
4
votes
1answer
242 views

Are these two definitions of exterior derivative equivalent?

I saw two definition of the exterior derivative of a $k$-form $\omega$. First definition: $$(d\omega)_p(v_0,\ldots,v_k)= \sum_{i=0}^k(-1)^id(\omega(v_0,\ldots,\hat{v_i},\ldots,v_k))_p(v_i)$$ Second ...
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0answers
97 views

Implicit function theorem to prove dimension invariance of diffeomorphisms?

Basic question. I started reading Ordinary Diff. Equations by V. I. Arnold and am a little confused about one of the exercises: proving a diffeomorphism from U to V (in the context of the text, this ...
2
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2answers
606 views

Composition of smooth maps

From section 1, problem 3 of Differential Topology by Guillemin and Pollack: Let $X \subset R^N, Y \subset R^M, Z \subset R^L$ be arbitrary subsets, and let $f : X \to Y, g : Y \to Z$ be smooth ...
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0answers
194 views

How to show that the diagonal of $X\times X$ is diffeomorphic to $X$?

The diagonal $Q$ in $X\times X$ is the set of points of the form $(x,x)$. Show that $Q$ is diffeomorphic to $X$, so $Q$ is a manifold if $X$ is. Can anyone please help me to solve this question I ...
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vote
2answers
71 views

Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
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vote
2answers
34 views

Differentiation along surface

I have this question I got when trying to solve a physics problem and I don't know which topic it belongs to. Please redirect me if anyone asked the same question before. I have a function ...
4
votes
1answer
281 views

Example of a nontrivial subbundle of a trivial vector bundle

It seems to me that you can have a nontrivial subbundle sitting inside a trivial vector bundle. Can anyone please give an example of this which one can visualize? Thanks!
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0answers
96 views

Continuity in the Strong(Whitney) Topology

Let $P,M$ and $N$ are smooth manifolds and let $F:P\times M \to N$ be a smooth map. We know its associated map $\tilde F:P \to C^\infty(M,N)$ given by $p \mapsto F_p(m)$ is continuous if and only if ...
8
votes
2answers
708 views

Let $G$ be a Lie group. Show that there is a diffeomorphism $TG \cong G \times T_e G$.

Since $T_p G$ is isomorphic to $T_e G$ for all $p\in G$, it makes sense that each vector in $T_p G$ can be identified with a vector in $T_e G$. Hence, to make the map from $TG$ one to one, we must ...
2
votes
1answer
67 views

set of critical values is measurable

I am reading John Milnor's Topology from a differentiable viewpoint. In Chapter 3 be proves Sard's theorem and claims (page 18) that if $g:R^n\to R^p$ is smooth with set of critical points $C'$ then ...
3
votes
2answers
160 views

Tangent Bundle of a Riemann Surface

Let $g$ be the genus of a closed Riemann surface, what can be said about $g$ if the tangent bundle $T$ of that surface is trivial? From the formula for the degree of a tangent bundle, $\deg(T)=2-2g$, ...
3
votes
1answer
93 views

Some questions about $S^n$

I have some questions about the $n$-sphere: I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it's true, but why is it not the case for other $n$? I have the same question for ...
0
votes
1answer
98 views

Bounded vector field on a closed surface

Let $S\subseteq\mathbb{R}^3$ a closed surface and let $X\in\mathfrak{X} (S)$ a vector field on $S$ such that $\mid\mid X_p\mid\mid \le M$ $\forall p\in S$ for some constant $M>0$. Prove that $X$ ...
4
votes
2answers
285 views

Describe tangent and normal bundle to a manifold

Consider the set $X:=\{(x,y,z)\in\mathbb{R}^3 | x^2+3y^2=1+z^2\}$ I have to show that $X$ is a submanifold of $\mathbb{R}^3$ (and this is trivial); then, using on $T\mathbb{R}^3$ the standard ...
0
votes
1answer
99 views

Explicitly Proving a parametrization for $x^2 + y^2 - z^2 = a$ for $a < 0$ is a Diffeomorphism.

Problem: I'd like to parametrize the manifold given by $\{(x,y,z)\in{\mathbb R}^{3}\,|\, x^2 + y^2 - z^2 = a\}$ for $a < 0$. The two mappings we'd use are $f(x,y) = (x,y,\sqrt{x^2 + y^2 - a})$ and ...
3
votes
2answers
749 views

Divergence theorem on Hyperbolic space

Given a vector field, say $F$, defined on a manifold $U$, the divergence theorem states that: $$\int_U\nabla \cdot F dV=\int_{\partial U} F d \Sigma .$$ Well if the manifold is $\mathbb R ^n$ and $F$ ...
0
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1answer
105 views

Relatively compact subsets of a manifold.

So I'm going through Otto Forster's "Lectures on Riemann Surfaces", and I need another hint (shame). This is in the "Cohomology Groups" sections, as part of a problem to show that for $X$ a compact ...
5
votes
2answers
337 views

Elementary Morse theory

I want to initiate myself to 'elementary' Morse theory and use it to calculate the Euler-Poincare characteristic of some compact manifolds (spheres and torus ...). I do not know what strategy should I ...
5
votes
1answer
152 views

Cancelling 3-handles in Kirby diagrams

Recently been trying to understand the proofs of Gompf and Akbulut that certain 4-manifolds are $S^4$ (these 2 papers: Gompfs paper in Topology Vol. 30 Issue. 1, Akbulut). In which they use a clever 2 ...
4
votes
1answer
201 views

Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book)

So I was looking at the proof given in Bott, Tu "Differential Forms in Algebraic Topology" of how to approximate continuous mapping by smooth mappings between manifolds. It is Proposition 17.8 on Page ...
1
vote
1answer
135 views

How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
4
votes
1answer
156 views

Stiefel-Whitney classes and lifts of structure groups

Let $M$ be a compact, smooth Riemannian manifold with tangent bundle $TM$. I will not distinct between $TM$ and the associated $O(n)$-frame bundle. I believe the following statements are true, but if ...
3
votes
0answers
64 views

Is a solution to some partially differential equation homeomorphic (or diffeomorphic) to a solution of an equation with a different covariance group?

Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
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votes
2answers
183 views

Structure of a $ C^{\infty} $-manifold

I was studying differentiable manifolds (an introduction) and found the following example, but I am confused. Example The function \begin{align} f: &\mathbb{R}^{3} \to \mathbb{R}, \\ f: ...
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votes
0answers
94 views

Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
25
votes
3answers
1k views

Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
1
vote
1answer
556 views

where can I find solutions to A comprehensive introduction to differential geometry by Spivak?

I have tried google and I fail to find solutions to the exercises in the book A comprehensive Introduction to differenial geometry volume I by Spivak. Does anyone know about a site with solutions to ...
5
votes
0answers
75 views

how to cover a manifold almost everywhere?

I pose the following question: Given a connected (maybe we need compactness) manifold or regular surface, can we find a single parametrization $\chi:B\to M$ from the open ball to the manifold, such ...
2
votes
0answers
313 views

Surfaces are homeomorphic iff are diffeomorphic.

I have read this statement in several places: "Two surfaces are homeomorphic iff are diffeomorphic". I think the nontrivial implication follows in this manner: First, we triangulate the surface and ...
0
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0answers
88 views

Why should the tangent bundle of the boundary of a conctractible manifold be stably trivial?

the question is already clear from the title, but I have to add at least 30 useless characters. The question is equivalent to ask if the normal bundle of the boundary is stably trivial
5
votes
1answer
141 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...