Tagged Questions

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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0answers
23 views

$C^{\infty}$-homotopy type of the Moebius band

The Moebius band $N$ has the same $C^{\infty}$-homotopy type of $S^1 \times \mathbb{R}$. What is the explicit expression of the $2$ $C^{\infty}$-homotopies involved ?
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0answers
13 views

Laplace-like operators on metric manifolds

i would like some help to understand the difference (and application) between the laplace-beltrami operator on a metric manifold (1) i.e and this form of a laplace-like operator (as an inner ...
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0answers
31 views

Derivations on a Manifold

Let $M\subset \mathbb{R}^n$ be an $m$-dimensional manifold (in the ordinary Euclidean sense). Given a point $p\in M$ we define a derivation $D_p$ (at $p$) to be linear functional on the space of ...
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0answers
23 views

A surface in $\mathbb{R}^{4}$ diffeomorphic to $\mathbb{S}^{2}$

This is homework so no answers please. The problem is: Show that the surface S given by \begin{matrix}x^{2}=-y\\ 1=y^{2}+s^{2}+t^{2}\end{matrix} is diffeomorphic to $\mathbb{S}^{2}$ My attempt: ...
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1answer
98 views

What is the space that we live in? [on hold]

Not sure if this question is trivial to some experts; but what is the three dimensional space that we live in? If this question is too difficult to describe, can we at least tell its topology? Is it ...
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1answer
16 views

Free and proper action

I don't know how to solve this problem. Let G be a Lie group and H a closed Lie subgroup ,that is, a subgroup of G which is also a closed submanifold of G. Show that the action of H in G defined by ...
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0answers
42 views

$x^{4}+y^{2}+z^{2}=1$ diffeomorphic to 2-sphere $\mathbb{S}^{2}$

This is homework so no answers please The problem is: $A=\{(x,y,z)\in \mathbb{R}^{3}: x^{4}+y^{2}+z^{2}=1\}$ is diffeomorphic to 2-sphere $\mathbb{S}^{2}$. Any mistakes: Consider ...
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0answers
25 views

The unit tangent bundle for submanifold $M^{m}\subset \mathbb{R}^{n}$ is a (2m-1)-dim submanifold

This is homework so no answers please Here is the problem: Show that $UM:=\{(x,v)\in T\mathbb{R}^{n}:x\in M^{m}, v\in T_{x}M^{m},|v|=1\}$ is a (2m-1)-dim submanifold of $T\mathbb{R}^{n}$. My ...
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0answers
8 views

Using the results of the local immersion/submersion theorems on manifolds

When $X,Y$ are $k$- and $l$-manifolds, we can have a function $f:X\rightarrow Y, x\in X$ such that $f$ is an immersion resp. submersion at $x$. The local immersion/submersion theorem now says: There ...
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0answers
30 views

Space of closed parametric curve is a manifold?

I have a problem that I need help to prove, can anyone please suggest any proof? Suppose we have, closed parametric curve $f(t)=(x(t),y(t))'$ for $t\in (0,2\pi)$ (here, map is ...
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0answers
20 views

What's the significance to the $m$ in the notation $L(n,m)$ for the Lens space?

I'm reading a quick example (Example 12.13 of Topological Manifolds by John Lee) of the construction of the lens space $L(n,m)$. Basically, let $$S^3=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^2+|z_2|^2=1\}$$ ...
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0answers
18 views

Relations between measure of the image set of a function between manifolds and its rank

Given $M^m$, $N^n$ smooth manifolds with $\dim M=m > \dim N =n$. $f\colon M \to N$ $C^1$ of rank $k < n$. Prove that $f(M)$ is a null set. my attempt We can't use Sard Theorem, because the ...
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1answer
21 views

Subspace not open of a differentiable manifold

Suppose $M$ is an orientable differentiable manifold with dimension $n$. $U$ is a subspace of $M$. If $U$ is not open, is it true that $U$ also is an orientable differentiable manifold ? I need a ...
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1answer
35 views

Proper and free action of a discrete group

In Gallot, Hulin, Lafontaine's Riemannian Geometry: Definition Let $G$ be a discrete group, acting continuously on the left on a locally compact topological space $E$. One says that $G$ acts ...
2
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1answer
35 views

Not all finitely-presented groups are fundamental groups of closed 3-manifolds

It is a well-known result that, for any finitely-presented group $G$ and any integer $n \geq 4$, there exists a closed $n$-manifold whose fundamental group is isomorphic to $G$ (a sketch of proof can ...
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0answers
9 views

Stereographic projection to show $S^n$ is a submanifold of $\Bbb R^{n+1}$

So $S^n$ in $\Bbb R^{n+1}$ can be described by the equation $x_1^2+\ldots+x_{n+1}^2=1$. Now consider two subsets $U_N:=S^n-\{(0,0,\ldots,1)\}$ and $U_S:=S^n-\{(0,0,\ldots,-1)\}$, the sphere less it's ...
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0answers
35 views

Preimage of the set of critical value of an analytic function between smooth manifolds

I have some problems with the following exercise, maybe due to alack of knowledge: Let $M$ be a connected smooth manifold and let $$ f \colon M \to N$$ be an analytic map. Denote by $C_f \subset M$ ...
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0answers
18 views

Embedding vs embedding as a closed subset

A version of Whitney's Theorem state that any $n-$dimensional manifold can be embedded in $\mathbb{R}^{2n+1}$ as a closed subset. Is there an example of $n-$manifold which can be embedded in ...
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2answers
20 views

How to define distance between two functions in a non-linear space (example of non-linear space: shape space)?

Suppose I have two parametric circle $f_1=(acost,asint)$ and $f_2=(bcos t,bsint)$, $t\in(0,2\pi),a>0,b>0$, which lies in some non-linear space. Are there any way, how to define the ...
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0answers
63 views

Edited: Proper nonsingular smooth map between connected manifolds is a covering map

Can you help me with this problem? Thanks Let $f:M->N$ be a proper nonsingular smooth map between connected manifolds. Dim(M) = dim(N). Show f is a covering map. Edit: So here is what I have so ...
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1answer
42 views

Clarification of notion of proper group action.

In a course on differential manifolds and Lie groups, the following theorem was stated, though never proven: Let $M$ and $N$ be smooth manifolds, and suppose $G$ is a Lie group acting on $M$. If ...
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0answers
38 views

Tangent Bundles to manifolds

I am having trouble trying to visualize exactly what a tangent bundle to the klein bottle is spuposed to look like. Is it possible for one to decompose it as a direct sum of simpler bundles?
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2answers
47 views

Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ with $f(x,y) = xy$ and $M=f^{-1}({0})$. Show that: The set $M$ is not a submanifold.

Assignment: Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ with $f(x,y) = xy$ and $M=f^{-1}({0})$. Show that: The set $M$ is not a submanifold. I've been able to show that sets are submanifolds ...
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0answers
28 views

Using Regular Value Theorem to Show that $S^2$ is a $2$-dimensional Manifold.

If we let $F(x,y,z) = x^2+y^2+z^2-1$. Then we know that $DF = (2x,2y,2z)$. I am confused as to how I should show that $0$ is a regular value of this function. I think I am missing something very ...
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1answer
36 views

Showing a function is a manifold

I have just been introduced to the world of manifolds in my real analysis class, and I'm having some trouble really understanding what manifolds are and showing why they exist. I have been given the ...
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0answers
53 views

Easiest Book to start Manifolds.

Ok, so here is the thing. I do not like analysis,Calculus or topology much, nor are they my strong point, but next week I have an exam on manifolds which I have a course First time in my life. I have ...
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1answer
35 views

A question related to the topology of the level sets of a particular type of smooth functions $f:\mathbb{R}^2\to \mathbb{R}$.

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function without critical points; i.e. such that $\nabla f(x)\neq (0,0)$, for all $x\in\mathbb{R}^2$. Is it true or false that all the level curves of $f$ ...
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0answers
19 views

Problem solving strategies in differential topology

I was wondering if there is a bag of tricks somewhere for differential topology and smooth manifold problems just like there is for analysis by prof. Tao ...
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1answer
31 views

Statistical Inference and Manifolds

I have just begun approaching the connection between statistical inference and differencial geometry. If I got it correctly, one of the most fundamental concept regards the connection between a $ ...
5
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1answer
59 views

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable (Tensors, or k-linear forms)

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable. $f\in A_k(V)$ is decomposable if there exists a $a_1,...,a_k\in V^\wedge$ such that $f=a_1\wedge...\wedge a_k$ In this case "let ...
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0answers
34 views

Are there non-manifold objects in real world?

I'm a beginner in Computer Graphics and today, I encountered the concept of "manifold". And according to the brief interpretation in Wolfram MathWorld: (http://mathworld.wolfram.com/Manifold.html), ...
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1answer
85 views

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ - I've got the gist, not sure how to write

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ With $A_k(V)$ being the vector space of alternating k-tensors. for $f\in A_k(V)$ for some $v\in V$ we define ...
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0answers
47 views

Tangent bundles of smooth manifolds

Using the identity $T(M \times N) = T(M) \times T(N)$, it is easy to construct the tangent bundles for various smooth manifolds such as the n-dimensional sphere $S^{n}$. However, I could not figure ...
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1answer
43 views

Manifold projection to 2m+1 dimensional subspace is a manifold.

Let $M \subseteq \mathbb{R}^n$ be a m-dimensional manifold. Suppose $n>2m+1$. Show that there is a projection from $M$ to a (2m+1)-dimensional subspace of $\mathbb{R}^n$ so that the image is ...
2
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0answers
39 views

Operations on a smooth vector bundle

On a smooth vector bundle, one often defines addition and scalar multiplication to form a vector space. However, doesn't one need to show that these operations are smooth? Is this trivial or is there ...
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1answer
29 views

Whitney sum of smooth vector bundles

I was reading through Lee's smooth manifolds book, in his chapter on vector bundles. Upon reading about smooth vector bundles and its definition, I was wondering if the whitney sum of two smooth ...
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0answers
20 views

Embedding of manifolds of constant negative curvature

Consider the manifold of constant negative curvature $G=SL(2, R)/\Gamma$ where $\Gamma$ is such that $G$ is compact (I have no special constraint on $\Gamma$). I know that by the Whitney embedding ...
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1answer
27 views

How to find tangent cone in singular point?

How to find tangent cone in singular point of surface? For example, considering surface in $\mathbb{R}^3$ given by equation $x^2z=y^2$, what is it's tangent cone in the origin? UPD:By tangent cone ...
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0answers
28 views

Are stable manifold for gradient flows embedded submanifold?

Generally, the stable manifolds $W^s(p)$ of a diffeomorphism $\phi:M\to M$ is no embedded submanifold. The injective immersion $$ E^s:T_p^sM\to M $$ does not need to be a homeomorphism onto its image ...
6
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1answer
85 views

Differential in Lie groups

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group ...
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0answers
78 views

Why Shape space is manifold?

In Shape analysis, often shape is considered as continuous parametrized closed curve and the shape space as Hilbert Riemannian manifold. Can any one help me to understand, why the shape space is ...
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2answers
29 views

Charts for level set manifolds & Multiplication map $F(A,B)=AB$ from $O(n)\times O(n)\to O(n)$ is smooth

This is homework so no answers please We have Multiplication map $F:O(n)\times O(n)\to O(n)$ defined as $F(A,B)=AB$ $F:O(n)\times O(n)\to O(n)$, where $O(n)=\{A\in M(n\times n):AA^{t}=id\}$. The ...
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1answer
38 views

Atlases on the topological manifold $\mathbb R$

I have been trying to produce an example of two incompatible atlases on $\mathbb R$. But no success. Could someone help me please? All my example seem compatible. For example, $A_1 = \{((-\infty,1), ...
1
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1answer
67 views

Exponential map and distance on a Riemannian Manifold

I'm trying to solve an exercise which I thought at first seemed simple but I'm having some trouble pegging down the error term. The question is to prove that on a Riemannian manifold we have ...
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0answers
53 views

Klein bottle visualization, parameterization, and isotropic version

Suppose a bright glowing orange mobius strip appeared in space for just an instant and then disappeared, except for its glowing orange edge, which remains suspended motionless in space for a moment, ...
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1answer
40 views

Correspondence between one-parameter subgroups of G and TeG

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
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3answers
60 views

does a commutative diagram implies pull-back?

Let $\xi=(E,p,B),\xi'=(E',p',B')$ be fibre bundles. Let $f: B\to B'$, $\bar f: E\to E'$ be maps such that the diagram commutes $\require{AMScd}$ \begin{CD} E @>\displaystyle\bar f>> E'\\ ...
1
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1answer
42 views

John Lee's Intro to Smooth Manifolds Inverse Function Theorem

In John Lee's "Intro to Smooth Manifolds" Chapter 7, p 160, we have a proof of the inverse function theorem. Here, in the middle of the page we have $F_2 = DF(0)^{-1} \circ F$. Is $DF(0)^{-1}$ a ...
0
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0answers
15 views

Distance from polynomial to Linear Manifold

How to calculate the distance form the polynomial $a(t) = 1+t^3$ to the Linear Manifold $H$ in the vector space $M_3$ $ H= \{ f(t) \in M_3 | f(1)=-1 \} $
3
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1answer
45 views

Second (and higher) derivatives of maps between manifolds

I'm trying to understand derivatives of maps between manifolds, and specifically something I read in Dodson and Poston's Tensor Geometry. I'll try to provide as much background as I can for those ...