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

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0
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3answers
46 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
2answers
50 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\...
2
votes
1answer
49 views

Pipe-fitting conditions in 3D

Let's we have smooth (continuous and infinitely differential) curve $f(x(t), y(t), z(t)) = 0$ in 3D. Now I want to build a tube of diameter $D$ around it. Questions: What are the set of conditions ...
0
votes
1answer
32 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 ...
0
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1answer
40 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 $1$-$1$...
2
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0answers
34 views
+50

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
votes
1answer
18 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 ...
0
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0answers
25 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 ...
8
votes
1answer
112 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 ...
0
votes
0answers
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 ...
2
votes
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, ...
0
votes
0answers
17 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
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 ...
1
vote
1answer
62 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 ...
12
votes
4answers
1k views

Why do we need Hausdorff-ness in definition of topological manifold?

Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant. ...
5
votes
2answers
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 \...
3
votes
2answers
109 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 ...
0
votes
0answers
20 views

Infinite cylinder a sub manifold

Is an infinite cylinder $$C=\{(x,y,z)\in R^3 \,| \, x^2+y^2= R^2\}$$ a $k$-dimensional differentiable submanifold of $\mathbb R^n$? And if so, what is the dimension $k$? Some help on what I need to ...
0
votes
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 $M$ ...
2
votes
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-...
7
votes
0answers
144 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
votes
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
votes
0answers
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) \...
0
votes
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 ...
1
vote
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 ...
0
votes
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, $$\frac{\partial }{\partial \theta}=(-\sin\varphi\sin\theta,\sin\varphi\cos\theta,0)$$ $$\...
2
votes
1answer
313 views

Wedge Sum of Two Spheres Homotopy Equivalent to a Compact Manifold?

Let $X=S^2\vee S^2$ (wedge sum). The homology groups are $H_0(X,\mathbb{Z})= \mathbb{Z}$, $H_1(X,\mathbb{Z})= 0$, and $H_2(X,\mathbb{Z})= \mathbb{Z} \oplus\mathbb{Z}$. I can see that $X$ is not ...
0
votes
0answers
29 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 ...
1
vote
0answers
52 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 ...
10
votes
1answer
86 views

Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...
7
votes
0answers
113 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 ...
4
votes
1answer
543 views

an injective immersion between two compact manifold of same dimension

$f:M\rightarrow N$ be a injective immersion, where $M$ and $N$ are same dimensional manifold with out boundary, we need to show $f$ is a covering map. what I tried is, $df_x:T_x(M)\rightarrow T_{f(x)}...
1
vote
1answer
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 ...
0
votes
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. ...
1
vote
2answers
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 ...
2
votes
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 ...
1
vote
0answers
34 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 $\...
1
vote
0answers
48 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
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-\...
2
votes
1answer
53 views

Laplacian of a submanifold in an Euclidean space

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$ ($n<m$). Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. ...
0
votes
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
vote
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 ...
0
votes
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 $x$ ...
0
votes
1answer
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 ...
0
votes
0answers
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
1
vote
0answers
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 $f:V\rightarrow W$ ...
0
votes
1answer
32 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 ...
2
votes
0answers
61 views

Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold.

Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold. The boundary is defined by $z = \sqrt{x^2+y^2}$. I'm trying to think how to approach ...
1
vote
0answers
51 views

Surjectivity of the exponential map on SO(2n)/U(n)

Let $M:=SO(2n)/U(n)$ the homogeneous space of all orthogonal almost-complex structures on $\mathbb{R}^{2n}$. When $n=2$, it is known that $M$ is just the 2-sphere. 1) On the 2-sphere, the ...