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

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moment map coordinates in tours action

I am trying to understand the proof of lemma 3.1, in this paper In proof, they say that $g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0$ I don't understand first and second equality. Can someone explain for ...
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25 views

Line with two origins is a manifold but not Hausdorff

The line with two origins is $(\mathbb{R} \times \{0,1\})/\sim$ where $(x,0)\sim(x,1)$ for $x\neq 0$. I can see that it is not Hausdorff, since we cannot separate the points $(0,0)$ and $(0,1)$. ...
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33 views

Are 1 dimensional connected closed smooth manifolds diffeomorphic to the $S^1$? [on hold]

Are 1 dimensional connected compact smooth manifolds without boundary diffeomorphic to the $S^1$ ?
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1answer
29 views

Local coordinates on a product of two manifolds.

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. I think that a local coordinate on $X \times Y$ is $(U \times V, x_1 ...
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1answer
21 views

A Smooth map homotopic to a constant map

Q: Let $M^{k}$ be a smooth compact $k$-manifold and let $F:M \rightarrow S^{n}$ be a smooth map, where $n>k$. Prove that $F$ is homotopic to a constant map. Proof: Since $n>k$, by Sard's ...
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1answer
32 views

Topological structure of the Manifold valued functions

$M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is ...
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14 views

Proof of Triangulation Theorem for 1-Manifolds

While I am reading "Introduction to Topological Manifolds" by John M. Lee, I come to see the following paragraph in the proof of Theorem 5.10 pp. 102. Note that Int$\ e\cap\ $Int$\ e'$ is open in ...
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1answer
28 views

How to prove line bundle L is trivial if and only if its dual bundle us trivial?

How to prove line bundle L is trivial if and only if its dual bundle us trivial ?
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14 views

Classification of surface with 18-gon planar diagram

For starters, this is a problem from L. Christine Kinsey's "Topology of Surfaces." The problem is to classify the surface using cut and paste arguments on polygons. However, between my limited ...
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41 views

How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
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1answer
25 views

How to define the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$?

I read a paper and on page 9, the paragraph before Proposition 5, it is said that let the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$. How to ...
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1answer
38 views

Sobolev space on closed surfaces

I was wondering if anybody here knows how the Sobolev space $H^2(\mathbb{S}^2)$ is defined? I.e. I want to integrate on this space with respect to the surface measure, but since this not the canonical ...
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1answer
22 views

Is this subset of a smooth manifold, a submanifold?

I'm not much informed about manifold but I should answer some questions about it. Based on the definition I have written an answer for the following question but I feel there is something wrong with ...
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1answer
31 views

Whitney's Embedding

The Whitney embedding theorem says that any smooth manifold of dimension $n$ may be embedded in $R^{2n}$. I am just beginning to study differential geometry for application to physics (general ...
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3answers
143 views

Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?

Is there any answer of this question around basic theory of differentiable manifolds?
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1answer
49 views

Build a topological manifold starting from a set.

Suppose you are given a generic set $X$. There exist sufficient and non-trivial conditions that ensure the existence of a topology $\tau_X$ on X such that the topological space $(X,\tau_X)$ is a ...
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1answer
34 views

Hypersurface is not curved, normal vector

Let $S$ be a hypersurface in $\mathbb{R}^n$. Is there a simple way to say that $S$ is flat by describing the normal vectors on $S$? Like $S$ is flat if the normal vectors on $S$ are all identical..
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201 views

Are compact complete geodesics closed?

Let $(M,g)$ be a compact Riemannian manifold. Is there an example of a geodesic $c:\mathbb{R}\to M$ s.t. $c(\mathbb{R})$ is compact, $c$ is NOT periodic (i.e. be NOT a closed geodesic) ?
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68 views

If it looks like a solid torus, walks like a solid torus, and quacks like a solid torus, is it a solid torus?

If an orientable 3-manifold $M$ has boundary the torus $S^1\times S^1$ and deform retracts to a solid torus $S^1\times D^2$, is it necessarily homeomorphic to a solid torus? Equivalently, if the ...
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2answers
49 views

Smooth homotopy

Let $M,N$ be manifolds. Suppose that $f_0, f_1:M\stackrel{C^\infty}\to N$ are homotopic, i.e. there exists a continuous mapping $f:M\times[0,1]\to N$ s.t. $f(x,0)=f_0(x)$, $f(x,1)=f_1(x)$. Then is ...
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24 views

Calculating Integral Submanifolds

I have the vector fields $v_{1} = x \partial_y - y \partial_x + z \partial_w - w \partial_z$ and $v_{2} = z \partial_x - x \partial_z + w \partial_y - y \partial_w$ on $S^{3} \subset \mathbb{R}^4$. I ...
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1answer
31 views

0th-order differential operator vs. 1st-order differential operator on a vector bundle $(E, \pi,M)$

Consider a vector bundle $(E,\pi,M)$. A 0th-order differential operator on $E$ is a $C^\infty(M)$-linear endomorphism $\Gamma E\rightarrow\Gamma E$. $\Gamma E$ is the set of sections on $M$. A ...
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Intersection of invariant subsets of a local group action

I don't understand some facts about invariant subsets of a local group action. Basically (to save you reading definitions) local actions are germs of partial actions which in turn are just like ...
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9 views

how can I do an implicit integration by “picking” the right function?

Supposing I have a function $f(x,y) = y/x$. The relationship of $y$ and $x$ is determined through an implicit unknown function $g(y,x)=0$, so that there is a unique path from two points. How can I ...
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1answer
20 views

Immersion locally injective?

I was wondering whether any immersion is locally injective?-This definitely sounds natural and I guess that it is true, as the derivative of an immersion is globally injective. Thus, there cannot be ...
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1answer
37 views

What are the elements in $\Gamma(\Lambda^2 TM)$?

In the lecture notes, Proposition 1.19 on page 9, it is said that on every Poisson manifold there is a unique bivector field $\Pi \in \Gamma(\Lambda^2 TM)$ such that $$ \{f, g\} = \langle \Pi, df ...
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29 views

Fundamental Group of an n-dimensional manifold with finite k punctures

Problem: Show that $\pi_1(M\setminus${k points}$) = \pi_1(M)$, where $M$ is an n-dimensional manifold ($n\ge 3)$ and k is a positive integer. In class, we went over the proof for the above ...
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1answer
57 views

Infinite Genus Riemann Surfaces

I want to show that every infinite genus Riemann surface $M$ has a proper closed subset such that, $M^*\setminus E$ ($M^*$ is the one-point compactification of $M$) is connected and locally ...
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0answers
43 views

Distance function under diffeomorphism of manifolds

community, I am concerned with measuring distances of systems under diffeomorphisms. Concrety, I consider a smooth diffeomorphism $\varphi: M\rightarrow N$ from the smooth differentiable manifold ...
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1answer
43 views

Differentiable Manifold minus point not compact

Suppose $X$ is an $n$-dimensional for differentiable manifold for $n \geq 1$: in our definition this is a second countable Hausdorff space with a maximal differentiable atlas. If $p \in X$ is a ...
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179 views

Rank Theorem proof

Let $\phi: M \to N$ be an immersion from smooth manifold $M^m$ into $N^n$ ($\dim M = m$ and $\dim N = n$). Prove there exists smooth charts $(U,h)$ in $M$ with $p \in U$, $h(p) = 0$, and $(V,g)$ in ...
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45 views

Frobenius theorem for 2-plane fields on some open set in $\mathbb{R}^3$

I need help with this two part question. I am rather confused by it. let $f(x, y, z)$, $g(x, y, z)$ be smooth on $U \subset \mathbb{R}^3$ with $f^2 + g^2 > 0$ on $U$. Define the differential form ...
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1answer
38 views

Diffeomorphism between the Grassmannian manifolds $\mathbf{Gr}(n,k)$ and $\mathbf{Gr}(n,n-k)$.

This seems to be a common exercise question, however I am having trouble with it. The hint is to use a map that associates the k-plane to its orthogonal complement. But I have not been able to show ...
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1answer
24 views

Component formula for pullback of one forms

Should it not be $\Big(F'(x)v \Big)^j = \frac{\partial F^j}{\partial x^k }(x)v^k$? Then also how is $F^*dy^j = \frac{\partial F^j}{\partial x^i}dx^i$ derived? I cannot what $\beta_j$ has been ...
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1answer
53 views

On the definitions of $n$-manifold etc.

I'm doing 3rd year undergraduate geometry (an introductory subject), and we've been given formal definitions of terms like "$n$-manifold" and "smooth $n$-manifold." However, I tend to think about ...
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pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
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1answer
42 views

function of class C ^ 1 on manifolds

Let $M$ be a differentiable manifold with finite dimension $ m $. Let $ f:M\rightarrow M $ a function of class $C^1$. I have a doubt about what this implies (1) or (2): $x \in M \rightarrow D_xf ...
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1answer
49 views

Smooth map on submanifold

Is the following true? Let $M$ be a differential manifold and $f : M \to M$ be a smooth map. If $N$ is a submanifold of $M$ and $f(N)\subset N$ then the restriction $f|_N : N \to N$ is smooth.
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2answers
90 views

frustrating experience about differential geometry

I am felling rather frustrated now, after taking a long time to study differential geometry, but with little progress... Indeed my major is mainly numerical analysis. I am studying modern geometry, ...
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1answer
38 views

Is rectangle manifold with boundary

Is a closed rectangle a 2d manifold with boundary? It seems like the corners don't have neighborhoods homeomorphic to the Euclidean half space?
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1answer
48 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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0answers
17 views

norm of $\text{grad} f$ 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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52 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
34 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
80 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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77 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
28 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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1answer
21 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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24 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 ...
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51 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 ...