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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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.
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458 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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550 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 ...
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362 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 ...
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115 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 ...
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406 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 ...
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161 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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70 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 ...
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114 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 ...
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1answer
178 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 ...
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244 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 $v_i$ ...
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1answer
108 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 $f\in C^1(\bar \Omega)$, ...
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44 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 ...
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1answer
56 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} ...
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2answers
136 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 ...
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271 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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108 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 ...
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826 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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230 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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2answers
78 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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36 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 ...
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383 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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107 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 ...
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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 ...
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70 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 ...
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199 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, $\deg(T)=2-2g$, ...
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106 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 ...
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1answer
105 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 $X$ ...
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2answers
332 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 ...
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117 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
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877 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 $F$ ...
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1answer
135 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 ...
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412 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
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1answer
174 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 ...
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1answer
330 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 ...
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1answer
162 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
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1answer
173 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 ...
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76 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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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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115 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 ...
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3answers
2k 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 ...
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1answer
643 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 ...
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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 ...
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353 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 ...
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1answer
155 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 $F:M\rightarrow\overline M$ ...
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1answer
50 views

Pairwise Disjoint Balls on a Manifold

I am wondering if it is always possible to find disjoint sets on any manifold such that these sets are balls when mapped to their locally Euclidean space $such$ $that$ there are an infinite number of ...
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152 views

Integration on manifold, pullback

Define the maps $F_\pm:B^2\rightarrow S^2$, $(x,y)\rightarrow(x,y,\pm\sqrt{1-x^2-y^2})$. Prove that for any $\omega\in\Omega^2(S^2)$: $$\int_{S^2}\omega=\int_{B^2}F_+^*\omega-\int_{B_2}F_-^*\omega$$ ...
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224 views

Embedded surface in $\mathbb{R}^3$

Let $U \subseteq \mathbb{R}^2$ be an open set and let $\sigma : U \rightarrow \mathbb{R}^3$ be a parametrization of an oriented surface $S$ embedded in $\mathbb{R}^3$ whose unit normal in $\sigma ...
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1answer
314 views

control of the $C^{1}$ norm of a diffeomorphism

Let $\mathcal{E}$ be the set of smooth manifolds with boundary $E\subset \mathbb{R}^{3}$ which are perturbations of the unit ball whose volume $V$, diameter $d$ and area of the boundary $A$ satisfy: ...
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584 views

Closed not exact form on $\mathbb{R}^n\setminus\{0\}$

I'd like to construct a closed but not exact $n-1$-form $\omega$ on $\mathbb{R}^n\setminus\{0\}$ in analogy to the winding form: $$\frac{x~dy-y~dx}{x^2+y^2}$$ I think something like ...