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

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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, \...
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24 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, ...
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8 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 ...
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1answer
47 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,\...
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1answer
29 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 ...
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1answer
24 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?
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2answers
148 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 ...
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46 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} ...
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1answer
81 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 ...
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3answers
59 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 ...
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1answer
54 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\...
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1answer
42 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 $1$-$1$...
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1answer
28 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 ...
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1answer
34 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 ...
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1answer
35 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 ...
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118 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 ...
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2answers
63 views
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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 ...
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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 ...
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33 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 ...
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1answer
63 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 ...
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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 ...
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23 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 ...
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2answers
110 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 ...
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1answer
46 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-...
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74 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 \...
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2answers
70 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, ...
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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 ...
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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 $M$ ...
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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 ...
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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 ...
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38 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) \...
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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 ...
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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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114 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 ...
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31 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 ...
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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, $$\frac{\partial }{\partial \theta}=(-\sin\varphi\sin\theta,\sin\varphi\cos\theta,0)$$ $$\...
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1answer
23 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 ...
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44 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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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 $\...
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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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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 ...
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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 ...
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44 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-\...
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2answers
80 views

Why $I = [0,1]$ is a $1$-manifold and $I^2$ not?

I am stuck in this, I have no idea why! $[0,1]$ is a manifold with boundary, how to justify? Which are the charts? And how about $[0,1]^2?$ Why it is not a manifold? My definition of topological ...
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1answer
44 views

Useful Formulas Analysis on Manifold

This is just a reference request. Does anybody has or know a book with a short handed formulary for Calculus on Manifold. I could do it, but surely someone already have done it better than I would. ...
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28 views

Calculate the area of $\partial G$ of $G = (z > x^2 + y^2$ and $x^2 + y^2 + z^2 < z)$ in $\mathbb{R}^3$.

Calculate the area of $\partial G$ of $G = (z > x^2 + y^2$ and $x^2 + y^2 + z^2 < z)$ in $\mathbb{R}^3$. I'm not really sure how to approach this. I've tried using spherical coordinates but I ...
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1answer
37 views

Connectedness and dimension of a manifold

Let $S=\{(x,a_3 , a_2, a_1 , a_0) \in \mathbb R^5 : x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 =0\}$ I want to show that $S$ is a connected manifold, and find the dimension of $S$. It seems that each $x$ ...
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28 views

Applications of Banach's fixed point theorem on Differential Geometry

Does anyone know any simple application of Banach's fixed point theorem on Differential Geometry. I am looking for something involving manifolds.f
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50 views

Simple properties of wedge product [closed]

How to prove a) $\omega \wedge \eta =(-1)^{kl}\eta\wedge\omega, \omega$ is $k$-tensor and $\eta$ is $l$-tensor. b)$f^*(\omega \wedge \eta)=f^*(\omega)\wedge f^*(\eta)$ where $f:V\rightarrow W$ ...
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1answer
34 views

Pullback of the metric on $\mathbb S^n$ on $\mathbb R^n$.

Let $\varphi:\mathbb R^n\longrightarrow \mathbb S^n$ the inverse of the stereographic projection, i.e. $$\varphi(y)=\left(\frac{2y}{\|y\|^2+1},\frac{\|y\|^2-1}{\|y\|^2+1}\right).$$ What I'm trying to ...