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.

learn more… | top users | synonyms

2
votes
0answers
47 views

Diffeomorphism between $T$ (torus) and the cover $S^1 \times S^1$ - Question of the book '' Differential Topology '' of Guillemin and Pollack

Here's the question : ''The'' torus is the set of points in $\mathbb{R^3}$ at distance $b$ from the circle of radius a in the xy plane, where $0<b<a$. Prove that these tori are all diffeomorphic ...
0
votes
1answer
21 views

Closed geodesic minimizing properties

Considering closed geodesics on a compact manifold M of even dimension, what does it mean to say that a curve (any closed geodesic) is locally energy minimizing but not globally ? For simplicity, say ...
0
votes
0answers
23 views

Why gradient-like dynamical systems are special case of Morse-Smale systems?

I'm studying Morse Theory and my question is exactly as stated in the above title. I can't see how a gradient-like dynamical system could be considered as a Morse-Smale system? Thanks in advance for ...
0
votes
0answers
30 views

The inverse image of any regular value is a submanifold

Let $f:M^m\subset\mathbb R^k\to N^n$ be a smooth map between manifolds with $m>n$. Let $y\in N$ be a regular value for $f$. Let $x\in f^{-1}(y)$ and $L:\mathbb R^k\to\mathbb R^{m-n}$ be a linear ...
0
votes
0answers
15 views

Applications of $C^\ast$ algebras in differential topology

I was wondering if there were any useful ways $C^\ast$ algebras come into play within differential topology. I know this is a fairly broad question,so any type of input is applicable.
2
votes
3answers
37 views

Union of the two coordinate axes in $\mathbb{R^2}$ - From the book '' Differential Topology '' of Guillemin and Pollack

I have tried to solve the problem : Prove that the union of the two coordinate axes in $\mathbb{R^2}$ is not a manifold. Let $X = \{(x,y) \in \mathbb{R^2} : x=0~or~ y=0\}$ be the union of the two ...
1
vote
1answer
49 views

Topological question from the book '' Differential Topology '' of Guillemin and Pollack

Here's the question : A smooth bijective map of manifolds need not be a diffeomorphism. In fact, show that $$f:\mathbb{R^1}\rightarrow {R^1}$$ $$x\rightarrow f(x)=x^3,$$ is an example. I would like ...
0
votes
0answers
32 views

On the following of the question: every $k$-dimensional vector subspace $V$ of … [duplicate]

This is the continuity of this question I've created this question recently, but I didn't receive all the answers I hoped. Someone could explain me why is it that the problem of approach work? In the ...
2
votes
1answer
38 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
3
votes
0answers
31 views

Focal points of the parabola $y = x^2$ in $\mathbb{R}^2$. [closed]

Let $X$ be an $n-1$ dimensional submanifold of $\mathbb{R}^n$, a "hypersurface." A point in $\mathbb{R}^n$ is called a focal point of $X$ if it is a critical value of the normal bundle map $h: N(X) ...
4
votes
0answers
27 views

Immersion except at the origin. [closed]

Whitney showed that for maps of two-manifolds into $\mathbb{R}^3$, a typical cross cap looks like the map $(x, y) \to (x, xy, y^2)$. Prove that this is an immersion except at the origin.
4
votes
1answer
60 views

Is this statement about manifold true? [duplicate]

Suppose $M$ is a closed $k-$manifold in $\mathbb R^n$ without boundary, can we always find a smooth function $f:\mathbb R^n\to\mathbb R^{n-k}$ such that $M$ is the level set where $f=0$?
5
votes
0answers
53 views

is there a diffeomorphism with only finite orbits but of infinite order?

Note: after not receiving any answer for some time, I asked this in mathoverflow, and got an answer there. The Question: Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have ...
4
votes
2answers
179 views

How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
2
votes
1answer
58 views

Working with homomorphisms and de Rham cohomology.

Here’s my question: Let M be a connected, compact, orientable, smooth n-manifold ($ n \in \mathbb{N}_{\geq 2} $). Let V be a neighborhood of p diffeomorphic to $\mathbb{R}^n$ and let U = M \ {p}. ...
0
votes
0answers
27 views

Tangent space chart dependent or not?

I was wondering whether the definition of a tangent space is chart dependent or not? Cause to me it seems that they depend on the charts: Let $p \in M$ where $M$ is a k-dim smooth manifold and $\phi: ...
1
vote
0answers
34 views

Is the $C^0$-fine topology finer than the metric topology?

Let $C(E,F)$ be the set of continouos maps between metric spaces $E$ and $F$. Suppose we are given the $C^0$ fine topology and a metric topology on $C(E,F)$. We know that the fine topology is finer ...
5
votes
2answers
78 views

Finding the de Rham cohomology of an open subset of $ \Bbb{R}^{n} $ minus a point.

Here’s my question: Let $ n \in \mathbb{N}_{\geq 2} $. Suppose that $ U \subseteq \Bbb{R}^{n} $ is an open set and that $ x \in U $. Then show that $$ {H_{\text{dR}}^{n - 1}}(U \setminus \{ x ...
4
votes
1answer
41 views

Can $\mathbb{R}\mathbb{P}^2$ be embedded into an orientable 3-manifold?

We know that $\mathbb{R}\mathbb{P}^2$ cannot be embedded into $\mathbb{R}^3$, but is there an orientable 3-manifold where it is possible?
1
vote
1answer
27 views

Extending a diffeomorphism outside a compact set

I believe that the following statement is true: Let $U,V\subset \mathbb{R}^n$ be open sets, $K\subset U$ compact, and $\gamma:U\to V$ a diffeomorphism. Then there is a diffeomorphism ...
0
votes
1answer
30 views

how to find points where a k-form is nonvanishing.

for example, if we are given 2-form $\omega=2xdx\wedge dy+2ydy\wedge dz$, what are the points where the form vanishes? I can only think of points $(0,0,z)$, is it all? Additionally, if we have a form ...
2
votes
2answers
45 views

Show that every k-dimensional vector subspace V of $R^N$ is a manifold diffeomorphic to $R^k$.

I'm actually in a exercise of the book " Differential Topology " of Guillemin and Pollack. Show that every k-dimensional vector subspace $V$ of $R^N$ is a manifold diffeomorphic to $R^k$, and that ...
1
vote
0answers
34 views

Index of a zero of a normal vector field

Let $M$ be a manifold and $S\subset M$ a submanifold. Assume whatever regularity you need (smoothness, compactenss, orientability, empty boundary, proper embedding of $S$,... this question is ...
3
votes
1answer
51 views

Difference between Euler characteristics of a Riemann surfaces

Let $X$ be a compact connected Riemann surface of genus $g$. Let $U$ be the complement of $r$ points in $X$. The Euler characteristic of $X$ = $2-2g$. That I understand. But I'm confused about the ...
3
votes
1answer
58 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
0
votes
0answers
23 views

Immersion of punctured torus into Euclidean [duplicate]

(a) Show there is an immersion of the punctured torus $S^1\times S^1$ - {a point} into $R^2$. (b) generalized it to $T^n$ - {a point} into $R^n$ can you give concrete proof for these problem? ...
1
vote
1answer
26 views

For a positive definite quadratic form $f: R^n \rightarrow R$, $f^{-1}(x)$, for any $x>0$, is diffeomorphic to $S^{n-1 }$

How to show for a positive definite quadratic form $f: R^n \rightarrow R$, there exists $f^{-1}(x)$, for any $x>0$, is diffeomorphic to $S^{n-1 }$?
3
votes
1answer
49 views

What intuition do we have for a subalgebra of Lie to be abelian?

The motivation for my question comes from the definition of rank of a given globally symmetric space: it is based on the image of a maximal abelian subalgebra of a given algebra by the exponential ...
0
votes
1answer
25 views

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 ...
3
votes
1answer
68 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 ...
2
votes
1answer
40 views

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 ...
2
votes
0answers
53 views

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} ...
6
votes
2answers
176 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 ...
0
votes
0answers
22 views

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 ...
2
votes
1answer
44 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 $\Bbb{R}^2$? ...
3
votes
0answers
51 views

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 ...
3
votes
3answers
66 views

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 ...
1
vote
0answers
28 views

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. ...
2
votes
0answers
37 views

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 ...
5
votes
1answer
171 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 ...
0
votes
1answer
101 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 ...
2
votes
1answer
51 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 ...
0
votes
1answer
13 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 ...
1
vote
1answer
43 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 ...
5
votes
2answers
80 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 ...
2
votes
1answer
40 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. ...
2
votes
1answer
39 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 ...
1
vote
0answers
55 views

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 ...
34
votes
3answers
543 views

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 = ...
7
votes
0answers
70 views

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 ...