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

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3
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1answer
26 views

Obtaining embedding from geodesic

Suppose $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$. And suppose I know the induced Riemmanian-metric $g$ on $M$, which ...
1
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0answers
52 views

The relationship between dimension of a manifold and coordinate function

I am thinking about the intrinsic meaning (what this equation really means) about this equation. Suppose $\mathcal M$ is a smooth manifold embedded in $\mathcal R^d$, then for any $x \in \mathcal M$,...
2
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0answers
51 views

computational insight behind why connections fix the shape of surface

Based on a video lecture, I had some queries. If we just have a manifold [M-set,O-topology,A-atlas] say $S^2$, this manifold represents a football or a potato equally. But once we choose a connection $...
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0answers
35 views

references for concrete computations in Lie groups for abstract toplogical concepts

A Lie group is a smooth manifold whose tangent space at its origin is its Lie algebra. Taking an example for lie group such as SL(2), and due to above facts we should then be able to translate the ...
-2
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1answer
27 views

covering space, smooth manifold

Let $p:Y\to X$ be a covering space and $p^{-1}(x)$ countable for every $x\in X$. Task: Let $X$ be a smooth manifold. Show, that $Y$ has the structure of a smooth manifold, regarding this $p$ is ...
1
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1answer
32 views

smooth manifolds, equivalent statements

Let $X,Y$ be smooth manifolds. Show: A function $f:X\to Y$ is smooth, iff for every open $V\subseteq Y$ and every smooth function $g:V\to\mathbb{R}$ the composition $g\circ f: f^{-1}(V)\to\mathbb{R}$ ...
0
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0answers
26 views

Confusion with chain rule when proving statement about tangent plane to a point in a manifold

I'm trying to prove the following: If $f:\mathbb{R}^3 \to \mathbb{R}$ is a differentiable function, $a \in \mathbb{R}$ is a regular value of $f$ and $S=f^{-1}(a)$, then for all $p \in S$ the tangent ...
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0answers
27 views

Clarification for Manifold and Manifold with Boundary Definitions

I am reading Spivak's Calculus on Manifolds, and just need a bit of clarification when it comes to definitions. He defines a manifold as some space $M$ that satisfies: $(M)$: $\forall x \in M, \...
0
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1answer
42 views

Existence of a universal cover of a manifold.

Suppose $M$ is a manifold, topological or smooth etc. As a topological space $M$ is required to be primarily locally homeomorphic to $\Bbb R^n$, with some added things that don't come along with this, ...
0
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0answers
26 views

Need help understanding part of this proof about local coordinates for Legendrian manifold

I need help understanding this proof in this book here: Concretely, I do not understand why it is okay to assume that $S$ can be parameterized by $n$ variables. Sure, it's an $n$-dimensional ...
0
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1answer
53 views

Integration on a submanifold - Where is my mistake

Let $$M=\{(x,y,z)\in \mathbb R^3 \mid z=1-x^2-y^2, z>0\}$$ be a two-dimensional submanifold. Now I need to integrate $$f(x,y,z)=\sqrt{\frac{1}{4}+x^2+y^2}$$ on $M$. I have chosen $$\phi(\alpha,\...
1
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1answer
30 views

Do these assumptions on a mapping ensure it is a diffeomorphism?

$A\subset \mathbb{R}^m$ is an open and bounded set, $f:A\longrightarrow \mathbb{R}^n$, $m\leq n$, is injective, continuously differentiable and its Jacobian matrix has full rank on $A$. Does this ...
0
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1answer
30 views

How to prove that a space is not a differential manifold?

Given a box (the surface of a cubic) in R^3 space, can I give a smooth structure on it to make it a differential manifold?
3
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2answers
158 views

Reference of what metric can be placed on manifold?

I just read some conclusion that $T^2$ can't be placed metric with positive curvature at all points. I don't know why is so . And what book introduce about this ? I mean about what metric can be ...
3
votes
1answer
51 views

How to find the transition function for two overlapping charts of $\mathbb{R}P^2$?

The real 2-dim projective space $\mathbb{R}P^2$ can be covered by the following 3 sets of unoriented lines through the origin un $\mathbb{R}^3$: $ U_x \doteq $ { all lines not lying in the yz plane} ...
1
vote
1answer
84 views

Difference between a manifold and a sub-manifold of the same dimension?

I appologize in advance in case this is a very trivial issue and for any mistakes due to translating stuff from my German lecture notes to English ... A subset $M \subset \mathbb{R}^n$ is defined to ...
0
votes
3answers
66 views

How to find a parametrization for a torus?

I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$ where $0<r<R$. I know I need to compute the metric tensor and ...
1
vote
1answer
59 views

Why the differential of exponential map is the identity.

Let $M$ a manifold and $T_pM$ it's tangent plan at $p$. We defined \begin{align*} \exp_p:U_p\subset \Omega _p&\longrightarrow M\\ V&\longmapsto \gamma _V(1) \end{align*} where $\gamma _V:I_V\...
0
votes
1answer
45 views

Version of Invariance of Domain for n-manifolds

I am working on the following exercise from Lawson's Topology: A Geometric Approach: Apply Invariance of Domain (If $U$ is an open subset of $\mathbb{R}^n$ and $f:U\rightarrow\mathbb{R}^n$ is $1$-$1$...
0
votes
1answer
31 views

Need a very simple example of coordinate functions and parameterization of a manifold

This is a very simple question from introductory differential geometry. Suppose I have an 2-dimensional manifold $M^2$ that is, for simplicity, a subset of $\mathbb{R}^2$. Now suppose $(U,\phi)$ is a ...
1
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1answer
36 views

Is integration with respect to spherical measure equivalent to manifold integration over sphere?

Let $S$ be an $n$--sphere $\mathcal{S}^n(0,R)\subset\mathbb{R}^{n+1}$, $\Theta\subset\mathbb{R}^n$ an open subset and let $\phi:\Theta\subset\mathbb{R}^n\longrightarrow S\subset\mathbb{R}^{n+1}$ be a ...
0
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0answers
42 views

Good reference for partitions of unity?

I am reading about Sobolev Spaces and regularity theory of PDEs. The partition of unity lemma, as stated in Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations, is as ...
8
votes
1answer
123 views

What is this manifold?

As picture below ,it is a Mobius band with a cylinder crossing it .Let it be $\Omega$ . Obviously , $\partial \Omega$ is a circle. Now , what is $\Omega/\partial \Omega$ ( I mean glue the boundary to ...
2
votes
2answers
83 views

Distance function on complete Riemannian manifold.

Let $\left(M, g\right)$ be a complete Riemannian manifold. Let us fix a point $p \in M$ and consider the distance function $$ r(x) := \operatorname{dist}(x, p). $$ I would like to characterize the ...
0
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0answers
19 views

Differentiable sub manifolds and regular parametrization

Let $0<r<R$. Consider the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \mid (\sqrt{x^2+y^2}-R)^2+z^2=r^2\}.$$ How can I show that $T^2$ is a two-dimensional differentiable submanifold of $\mathbb ...
0
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0answers
36 views

integration by parts on hypersurfaces

Usually the integration by part on the surface is trivial for planar domains. However, when it comes to hypersurfaces, some other terms like curvature show up. Can someone help with the understanding ...
1
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1answer
68 views

Show that there exists no immersion f of S 1 into R^1

I'm studying Differential Forms and Applications by Manfredo P.do Carmo. First, I suppose that there exists f s.t. df is injective. I guess the problem can be solved to use Stoke's theorem and other ...
0
votes
1answer
32 views

Compact differentiable sub manifold with at least two points [closed]

Let $M$ be a differentiable submanifold of $\mathbb R^n$ which contains at least two points. How can I show that if $M$ is compact in $\mathbb R^n$ there exists no atlas for $M$ which only consists of ...
0
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0answers
25 views

Infinite cylinder a sub manifold

Is an infinite cylinder $$C=\{(x,y,z)\in \Bbb{R}^3 \,| \, x^2+y^2= R^2\}$$ a $k$-dimensional differentiable submanifold of $\mathbb R^n$? And if so, what is the dimension $k$? Some help on what I ...
3
votes
2answers
119 views

The integral of a function on manifold and differential form

When we want to integrate a function f over a manifold M, we may meet some problems, for example, the problem showed in the picture below: Then people used differential form to integrate. But it ...
2
votes
1answer
50 views

Lie bracket and inner product

$X, Y, N$ are vector fields on a riemannian manifold $M$. $\langle X, N \rangle(p)=0 $, $\langle Y, N \rangle(p)=0$ at some point $p$. Show $\langle [X,Y], N \rangle(p)=0$. I want to use $[X,Y]=XY-...
5
votes
2answers
80 views

Grassmanian $(2, 4)$ homeomorphic to $S^2 \times S^2$

Prove that the Grassmanian manifold $G(2, 4)$ of all real two-dimensional planes in $\mathbb{R}^4$ that pass through the origin is homeomorphic to the product of two two-dimensional spheres $S^2 \...
2
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2answers
71 views

Problem to conceptualize $\frac{\partial }{\partial \theta}$ and $\frac{\partial }{\partial \varphi}$.

I have some little problem to give a conception to $\frac{\partial }{\partial \theta}$ and $\frac{\partial }{\partial \varphi}$ on manifold (like $\frac{\partial }{\partial x}$ as well). For example, ...
0
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1answer
20 views

Component square integration on the unit $n$-sphere

I found (with some hints from a nice math.se user) numerically that $$\int_{S^{n}} x^2 dS = \frac{1}{n+1} \int_{S^{n}} dS$$ where $S^n$ is the unit $n$-sphere in $\mathbb{R}^{n+1}$ and $x$ of course ...
0
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1answer
64 views

A question about pullback of the K-form

Let $M$ be an oriented $m$-dimensional manifold. Suppose the support of $\omega$ is in an open subset $U$ of $M$, and $\phi \colon U \to R^m$, $\psi \colon U \to R^m$ are two different charts on $M$ ...
0
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0answers
18 views

Details for SE(3) being a manifold

As a student of engineering, i read that SE(3) is a manifold which commonly is known to us as a transformation matrix. I have read proofs showing that a sphere is a 2-dimensional manifold. The proof ...
7
votes
0answers
148 views

Sard's Theorem Using Integration?

How exactly do we clean up this heuristic proof of Sard's theorem, from Schwarz' `Differential Topology for Physicists' using the Jacobian $J(f)(x)$: "A heuristic justification for Sard's theorem ...
0
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0answers
42 views

On finding a second countable basis for the tangent bundle $TM$

Let $M$ be a manifold. I want to show that the tangent bundle $TM$ is second countable. I know that for a given chart $(U, \phi)$ on $M$ we have a homeomorphism $D_{\phi}$ between $TU$ and $\phi(U) \...
0
votes
0answers
30 views

Second derivative/tangential map on a product manifold

Short version: Given $f: M_1 \times M_2 \longrightarrow N$ what is the second tangential $TTf$ expressed in partial tangentials on $M_1, M_2$? The details: Let $M_1, M_2, N$ be manifolds. The partial ...
0
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0answers
82 views

Bundle isomorphism $\Phi : TM \oplus T^*M \to T(T^*M)$.

Prove that there is a bundle isomorphism $\Phi : TM \oplus T^*M \to T(T^*M)$ which identifies the summand $T^*M$ with the vertical vectors. If $\omega_{can}$ is the canonical symplectic ...
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0answers
118 views

Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
1
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1answer
36 views

Complete Vector field

I am reading "Geometry of Differential Forms". We want to show that on a smooth compact manifold, vector fields are complete. We claim that there is an interval $(-\epsilon ~ ~\epsilon)$ of time ...
0
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1answer
36 views

Compute Christoffel symbol of $\mathbb S^2$.

Let $$(x,y,z)=f(\theta,\gamma )=(\sin \varphi\cos\theta,\sin\varphi\sin\theta,\cos \varphi).$$ Therefore, $$\frac{\partial }{\partial \theta}=(-\sin\varphi\sin\theta,\sin\varphi\cos\theta,0)$$ $$\...
1
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1answer
30 views

Geodesics on $SO(n)$

I'm trying to prove the following exercise about the ortogonal symmetric group $SO(n)$. I have been able to prove the first two sections of the exercise but I got stuck on the third. I don't ...
1
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2answers
45 views

number of roots on SO(3)

Suppose we have a smooth map$ f:SO(3) → SO(3)$ of manifolds s.t.$ f(X)=X^2$. $I$ though since I is a regular value of this map and f is orientation preserving, to calculate degree of it, it is enough ...
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0answers
36 views

If for each $\alpha, \beta$, the map $\psi_{\beta} \circ F \circ \phi_{\alpha}^{-1}$ is smooth, then $F$ is smooth.

Let $F:M\to N$ be a map. Suppose $\mathcal{A} = \{(U_{\alpha},\phi_{\alpha})\}$ and $\mathcal{B} = \{(V_{\beta},\psi_{\beta})\}$ are smooth atlas for $M$ and $N$ respectively. Suppose that for each $\...
2
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1answer
33 views

Question about connections and usual derivative.

Let $\nabla $ a covariant derivative. What does mean "in the normal coordinate, $\nabla $ is equivalent to the usual derivative". I recall that the normal coordinate is coordinate system on a normal ...
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0answers
33 views

A version of the regular value theorem [duplicate]

Assuming the regular value theorem, let $$f : \mathbb{R}^n\times \mathbb{R}^k \to \mathbb{R}^n.$$ Let $N = \{ x \in \mathbb{R}^n : f^1(x) = \ldots = f^{n-1}(x) = 0, ~~ f_n(x) \ge 0\}$. Supposing ...
1
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0answers
49 views

Is this map an immersion?

Let $g:\mathbb{R}^2\to \mathbb{R}^4,\ (x,y)\mapsto ((2+3\cos(2\pi x))\cos(4\pi y),\ (2+3\cos(2\pi x))\sin (4\pi x),\ 3\sin(2\pi x)\cos(2\pi y),\ 3\sin(2\pi x)\sin(2\pi y))$ I have to prove that for ...
4
votes
1answer
46 views

Bianchi identity proof : why we can consider $[X,Y]=[Y,Z]=[X,Z]=0$?

I recall that the Riemann curvature tensor is defined by \begin{align*} R:\Gamma(M)\times \Gamma(M)\times \Gamma(M)&\longrightarrow \Gamma(M)\\ (X,Y,Z)&\longmapsto [\nabla _X,\nabla _Y]Z-\...