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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closure of a smooth 1-manifold

We can define a smooth 1-manifold by a single coordinate patch $\alpha(t)=(e^t\cos(t),e^t\sin(t))$,$t\in\mathbb R$. We can show it is a smooth manifold in ${\mathbb R}^2$, but not closed. The closure ...
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Unsolved regular value problem in $\mathbb{R}^n$

I want to show that if $F : \mathbb{R}^n \rightarrow \mathbb{R}^{n-k}$ is a $C^1$ function and $rank(DF) = n-k$ then $M:=F^{-1}(\{0\})$ defines a manifold. My idea: Without loss of generality I ...
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47 views

Open Unit Ball diffeomorphic to the Open Unit Cube

How can I show that the open unit cube $(-1,1)^n \subset \mathbb{R}^n$ and the open unit ball $B = \{x \in \mathbb{R}^n \mid \|x\| < 1\}$ are diffeomorphic? I know that one can proof this by ...
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A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph

The stable manifold theorem tell us: A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the ...
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Embed $S^{p} \times S^q$ in $S^d$?

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} ...
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161 views

Why would one care about Fibre Bundles

As a physics student I can easily understand the motivation for studying manifolds and why the definition looks the way it does, I only have to think of Minkowski space in GR. But for the life of me ...
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Induced bundles and Smooth Maps

Given a smooth map between compact manifolds without boundary what criteria guarantee that the induced bundle is isomorphic to the tangent bundle? And less generally, suppose the map has non-zero ...
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1answer
31 views

immersion of punctured torus in plane

Let $S^1 \times S^1$ be the $2$-torus. If a point $a=(p,q)$ of the torus is removed, i.e., it is punctured at one point then how can I show that it can be immersed in the plane, i.e., in $\Bbb{R}^2$? ...
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Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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The biggest degree of a map between fixed surfaces

Let $S_1$ and $S_2$ be compact surfaces (real manifolds of dimension 2). None of them is a sphere. Question: What is the biggest degree of a smooth map from $S_1$ to $S_2$? Comment 1. I have a ...
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On critical values of a linear function $g:SO_2\times \mathbb R^2\to \mathbb R^2$

Consider map $g:SO_2\times \mathbb R^2\to \mathbb R^2$, $g(A,v)=Av$, where $A\in SO_2$ is an orthogonal $2\times 2$ matrix and $v\in \mathbb R^2$ is a $2$-vector. Show that $0$ is a critical value. ...
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How to prove that the tangent map $T\phi$ into the pullback bundle is smooth?

Assume $\phi: M\rightarrow N$ is smooth. Let $\phi^*(TN)$ be the pullback bundle of $TN$ by $\phi$. Define $T\phi:TM\rightarrow \phi^*(TN)$ as follows: $T\phi(m,v)=(m,d\phi_m(v)) $. We also have the ...
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165 views

Different definitions of differential forms?

I am a physicist and was reading about differential forms in Classical Mechanics. Now, I thought that a two-form is a smooth map $\omega : M \rightarrow \Lambda(T^*M)$ so that a point $p$ on the ...
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1answer
95 views

Differential forms - looking for 3 definitions!

I am sorry for this type of question, but I currently have to deal with differential forms although I have not heard so far what they actually are, so I have just a few very particular questions about ...
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1answer
45 views

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$.

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$. I've been looking for an diffeomorphism between a sphere in $\mathbb R^3$ and an ellipsoid of the form $$\{ (x,y,z) \in \mathbb R^3 ...
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1answer
12 views

How do I see that a homeomorphism $\sigma$ is an open function?

How do I see that a homeomorphism is an open function ? Given a homomorphism $\sigma: X \rightarrow Y$ between topological spaces, how do i then see that $\sigma(V)$ is open in $Y$ for $V$ open in ...
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1answer
33 views

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface?

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface ? Consider a regular surface $S \subset \mathbb R^3$ and a diffeomorphism ...
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2answers
75 views

What are monomorphisms in the category of real vector bundles over a fixed base space $X$?

I have a (perhaps stupid) question dealing with vector bundles. In most textbooks, a morphism of real vector bundles $f:\xi\to \eta$ over a fixed base space $X$ is said to be a ...
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1answer
37 views

Conifolds and Exotic Spheres

First of all, a disclaimer: I'm a physicist trying to understand mathematical aspects of some solutions I've encountered while studying string theory, and am certainly not a mathematician of any sort. ...
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1answer
34 views

What does “orthogonal complement” mean in Milnor's Topology from the Differentiable Viewpoint?

Milnor writes on p. 11 If $M'$ is a manifold which is contained in $M$, it has already been noted that $TM'_x$ is a subspace of $TM_x$ for $x \in M'$. The orthogonal complement of $TM'_x$ in ...
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a question of my real analysis class,can someone help me solve this question?

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
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Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = ...
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Is a compact, simply-connected 3-manifold necessarily $S^3$ with $B^3$'s removed?

Let $M$ be a compact, simply-connected 3-manifold (which is also smooth and connected). Is $M$ diffeomorphic to $S^3$ with a finite number of $B^3$'s removed? This seems like a handy fact, but I ...
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39 views

A question about the index of vector field

$M$ is the boundary of a compact manifold $U$: $M = \partial{U}$, $\mathbf{v}$ is a unit vector field on $M$, how to prove that if $\mathbf{v}$ can be extended to be a nonvanishing vector field on all ...
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1answer
30 views

Interior of image of regular points is dense?

I'm studying some problems related to differential topology and I came across the following exercise: if $f:M\rightarrow N$ is a surjective smooth (i.e., $C^\infty)$ function, $\dim(M)>\dim(N)$ and ...
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Does taking the derivative with respect to vector fields commute with taking submnaifolds?

Let $M$ be a smooth manifold and $N$ a submanifold of M. Let $X_1,..,X_k\in\Gamma(TM)$ be vector fields on $M$, which restrict to vectorfields on $N$, i.e. for $n\in N$ it holds $X_{i,n}\in T_n ...
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Modifying a smooth function with respect to conditions on its partial derivates

Let $\{U_i\}_{i\in I}$ be a locally finite collection of open subsets of $\mathbb{R}^n$, $K_i\subseteq U_i$ compact subsets, $\epsilon_i>0$ positive real numbers and a nonnegative natural number ...
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44 views

Guillemin & Pollack's proof on Whitney embedding theorem

I am confused with a little detail in Guillemin & Pollack's proof on Whitney embedding theorem. Please see page 54 in their book "Differential topology". In the second paragraph of page 54, they ...
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Ambiguity in definition of $C^r$ maps between manifolds

Let $M$ and $N$ be smooth manifolds with corresponding maximal atlases $A_M$ and $A_N$. We say that a map $f : M \to N$ is of class $C^r$ (or $r$-times continuously differentiable) at $p \in M$ ...
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Passage in a proof of a lemma

Here is a lemma and a proof given to me in class. Lemma If $M$ is a smooth manifold, $K\subseteq M$ a compact subset, $A\subset M$ an open set containing $K$< then there exists a compact-support ...
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211 views

Why can we always take the zero section of a vector bundle?

$\require{AMScd}$ As I understand it, a rank $k$ vector bundle is a pair of topological spaces with a map between them $$ E\xrightarrow{p}B $$ such that there exists an open cover $(U_\alpha)$ of $B$ ...
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1answer
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Are transverse maps in intersection theory local diffeomorphisms?

Suppose that $f: X \rightarrow Y$ and $Z$ is a submanifold of $Y$, all boundaryless. Suppose that $f$ is transverse to $Z$, so that: $$df_x T_xX + T_{f(x)}Z = T_{f(x)}Y$$ for every $x \in f^{-1}(Z)$ ...
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$SL(3,\mathbb{R})$ is a smooth manifold?

How do you show $SL(3,\mathbb{R})$ is a smooth manifold? I am thinking to use the preimage theorem, but what kind of thing I need to show first before I can apply the theorem?
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Intuitive meaning of immersions

I have a hard time understanding the concept of immersions. In my course, it was only introduced by the immersion theorem wich says: Let $f: U \subset \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ be ...
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71 views

Derivative of determinant at some point

Let $c:\mathbb{R} \rightarrow \mathbb{M}_n(\mathbb{R})$ defined by $$c(t)=A e^{tB}$$ where $A\in GL(n,\mathbb{R})$ and $B \in \mathbb{M}_n(\mathbb{R})$. The question ask me to find $c'(0)$ and ...
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1answer
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Want to show two smooth manifolds are diffeomorphic

Consider a smooth manifold $M = \{ (u,v) \in \mathbb{R^3} \times \mathbb{R^3} \mid \|u\|=\|v\|=1 \text{ with } u \perp v \}$, and want to show $M$ is diffeomorphic to $SO(3)$, the rotational group in ...
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1answer
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Are open sets in $R^n$ homeomorphic to $R^n$?

I am working on exercise 1.1 and I think the way to do this would be to show that open sets are homeomorphic to $R^n$ or open balls in $R^n$. Is this even true? I'm not sure how to go about ...
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a question about finding the points where Df(x) (derivative of f) is an isomorphism.

Let E be the four-dimensional real vector space $M_{2\times 2}$ of real 2$\times$2 matrices. Show that by setting f(X)=X^2 for 2$\times$2 matix X,we define a continously differentiable function f ...
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Construction of a diffeomorphism handling varying domain

Let $\Omega$ be a strictly convex domain, $\partial\Omega\in C^{2,1}$ We define a foliation $\{\Omega_t\}_{0\leq t\leq 1}\subset\Omega$ as follows. Let $\Omega_0=B_r(x_0)$, a small ball centered at ...
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Manifolds with smooth structure

One of the remark in my lecture notes said: In dimension $\leq 3$, every topological manifold has a unique smooth structure (up to diffeomorphism.) I don't quite understand what is a structure ...
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Prove each coordinate is a differentiable function

This is an exercise from a book called "Differential Topology" 2-11: Let $M$ be the sphere $x^2+y^2+z^2=1$ in 3-space. Prove that each of the Euclidean coordinates $x,y,z$ is a differentiable ...
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freedom in choosing a smooth function of compact support

Suppose $\Omega$ represents a bounded domain in $\mathbb{R}^n$. For ease, let $\Omega$ be a ball around the origin. Let $\varphi$ be a smooth radial function defined on $\Omega$. I am trying to ...
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114 views

Explain densities to me please!

When it comes to integration on manifolds, I speak two languages. The first is of course the language of differential forms, which is something I am relatively well acquinted with. The second ...
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1answer
53 views

Isometry of two Euclidean structures on the same vector bundle

I'm reading Milnor & Stasheff's Characteristic Classes, and I was struck by the following problem, which they call the Isometry Theorem: "Let $\mu$ and $\mu'$ be two different Euclidean metrics ...
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Manifolds and CW-complexes [duplicate]

This is a very naive question. Every manifold (assumed to be paracompact) is a CW-complex? Thanks.
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Connection between Chladni Plates and Algebraic Topology?

Does anybody know of a connection between Chladni Plates and Algebraic Topology? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
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The tangent space of a manifold in some given point.

My question is about the tangent space of a manifold in some given point. Let $M$ be a differential manifold and $(U,\varphi)$ a chart around a given point $p$ of $M$ . My question is : Is that the ...
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Constructing complex line bundles on orientable smooth manifolds

This questions ask how to construct a complex line bundle over a smooth compact orientable manifold without boundary starting with an n-2 dimensional orientable sub manifold without boundary. The ...
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63 views

Transverse submanifolds in product manifolds.

Suppose we have smooth manifolds $M,M',N$, a smooth map $f\colon M\rightarrow M'$ and a smooth submanifold $S'\subseteq M'\times N$, such that the projection $\pi_{M'}\colon S'\rightarrow M'$ is a ...
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103 views

Embedd the Klein bottle into a 3-manifold

Can the Klein bottle $K$ be embedded into $S^{2} \times S^{1}$? If can, how it works. If not, is there an obstruction? Thanks in advance.