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

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1answer
99 views

Meaning of the adjoint representation of a Lie group

The adjoint representaion of $G$ is a homomorphism $ad_{a}:g \rightarrow aga^{-1}$, $a,g \in G$, what is the meaning of this? Now if we identify $T_{e}G$ with $\mathfrak{g}$ we have the adjoint map ...
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1answer
122 views

Norm of the gradient of function $f$ on manifold $g(x)=c$

Let $g,f:\mathbb{R}^2 \to \mathbb{R}$, $M=g^{-1}(c)$. Let say that we manage to write $f(x,y)=f_{*}(x)$ for $x\in M$. When I was calculating the square of norm of $$\nabla f_M (x,y)=\nabla ...
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1answer
97 views

Integrating tensors on manifolds

When/how can you integrate tensors on manifolds and what does it mean? I imagine that line integrals of tensors make sense when you have a connection, since you can uniquely parallel transport all ...
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2answers
42 views

How to show $X=\{p\in M: \textrm{ker}(df(p))=\{0\}\}$ is open in $M$?

Let $M$ and $N$ be two smooth manifolds and $f:M\longrightarrow N$ a $C^\infty$ map. We say $f$ is an immersion at $p\in M$ if $df(p):T_pM\longrightarrow T_{f(p)}N$ is injective. How can I show the ...
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1answer
100 views

One-forms in differentiable manifolds and differentials in calculus

Suppose that we have this metric and want to find null paths: $$ds^2=-dt^2+dx^2$$ We can easily treat $dt$ and $dx$ "like" differentials in calculus and obtain for $ds=0$ $$dx=\pm dt \to x=\pm t$$ ...
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3answers
102 views

Constructing a vector bundle using Vector bundle construction lemma

Given are: an open cover of $\{U_\alpha\}_{\alpha\in A}$ of a smooth manifold $M$. smooth maps $\tau_{\alpha\beta}\colon U_{\alpha}\cap U_{\beta}\rightarrow \text{GL}(k,\mathbb{R})$ with ...
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0answers
50 views

quaternion vector bundle and quaternion grassmannian

Let $\mathbb{H}$ be quaternion numbers. Let $G_n(\mathbb{H}^\infty)$ be the grassmannian of $n$-subspaces of $\mathbb{H}^\infty$. Then ...
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2answers
72 views

Why does this Jacobian have full rank?

I am doing a basic exercises and I have to show that the set of $n\times n$ orthogonal matrices form a manifold. Naturally, I have defined a function $f(X) = X^TX-I$ and am considering $f^{-1}(0)$. ...
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1answer
26 views

Dimension of a sphere generated by rotation matrices applied to a constant matrix

One can easily check that for a given vector $\gamma$, $U\gamma$ for all the $U$'s which are rotation matrices defines a sphere with the Euclidean metric. For a given matrix $A\in M_{m\times ...
2
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1answer
147 views

Surface has Euler characteristic 2 iff equal to sphere

Let $\Sigma$ be a connected (not necessarily compact) surface with or without boundary. Is it true that $\Sigma$ is homeomorphic to the sphere if it has euler characteristic $\chi(\Sigma)\geq 2$? I ...
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2answers
59 views

Definiton of Submanifold of Topological Manifold

Analogy to smooth manifold, I want to define the submanifold of topological manifold. There are two ways. Let $M$ be a topological manifold, and $N\subset M$. If $N$ is a topological manifold, then ...
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1answer
38 views

Relation between Map and Dimension

I am curious about two questions below Let $M$, $N$ be two topological manifold. If $\dim M>\dim N$, is there exist an injective continuous map $f: M\rightarrow N$? If $\dim M<\dim N$, is ...
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0answers
27 views

what is the basis for $A_3(T_p(\mathbb{R^4}))$.

Let $x^1,x^2,x^3,x^4 $ be the coordinates on $\mathbb{R^4}$ and $p$ a point in $\mathbb{R^4}$. Write down a basis for the vector space $A_3(T_p(\mathbb{R^4}))$. I have no idea what is the basis for ...
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1answer
42 views

Vector field notation - what is the law of multiplication?

You are given three smooth vector fields on a differentiable manifold $M$. Take the Lie bracket: $[X,Y]f=X(Yf)-Y(Xf)$ My question is what is the law of multiplication between $X$ and $Y$? ...
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0answers
72 views

Why is this map of sheaves surjective?

Let $M$ be a complex manifold. Let $\mathcal{L}$ be a holomorphic line bundle over $M$, with $\pi:M\longrightarrow\mathcal{L}$ being the projection map. Any section $s$ of $\mathcal{L}$ over $M$ is a ...
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1answer
51 views

differentiation on manifolds(?)

$f=f(x_1,…,x_n)$ and $g=g(x_1,…,x_n)$ are two differentiable functions. $\frac{\partial f}{\partial x_1} = \frac{df}{dx_1}\big|_{x_2,…x_n}$ is the partial derivation with respect to $x_1$ while ...
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0answers
19 views

Is this choice of coordinates suitable for R?

In an exercise, I found the manifold $M=\mathbb{R}$ with the coordinate function $\phi(x)=x^2$. Since in the trivial topology the union of all $(U_i,\phi_i)$ doesn't give M, is this choice of ...
3
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1answer
151 views

How to calculate differential forms on $S^2$ parameterized by stereographic projection?

Suppose that we have the stereographic mapping $\varphi: \mathbb{R}^2\to M$ where $M=S^2-\{(0,0,1)\}$. I've already found that the stereographic parametrization of $S^2-\{(0,0,1)\}$ is given by: ...
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0answers
46 views

Compute the Jacobian of the following

The question is this: Consider the parabaloid $\{(x,y,z)|z=1-x^2-y^2\}$, let $A$ be the subset satisfying $z>0$. Consider the plane $\pi$ given by $z=1$. The functions $x$ and $y$ act as ...
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0answers
42 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
100 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
66 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
37 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
113 views

What is the space that we live in? [closed]

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
30 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 ...
2
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0answers
69 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 ...
3
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0answers
91 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
52 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
49 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
33 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
24 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
26 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
116 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 ...
3
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1answer
105 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
43 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
58 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
82 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. Another version states that that any $n-$dimensional manifold can be ...
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2answers
54 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
86 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
262 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
51 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
61 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 ...
3
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0answers
48 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
62 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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1answer
56 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
37 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 ...
1
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1answer
50 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 $ ...
7
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1answer
109 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
41 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), ...
6
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1answer
106 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 ...