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

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1answer
42 views

Extension of Smooth Functions on Embedded Submanifolds

In Lee Smooth Manifolds, this problem is given: if $S \subset M$ is smoothly embedded and every $f \in \mathcal{C}^{\infty}(S)$ extends to a smooth functional on $\textit{all}$ of $M$, then $S$ is ...
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0answers
34 views

Lie Algebra of $\mathrm{SO}(2)$ and $\mathrm{O}(2)$ are the SAME - why?

If $G$ is a Lie Group (with identity element of $e$), then my definition of the Lie Algebra $\mathfrak{g}$ of $G$ is the tangent space of $G$ at $e$, so that $\mathfrak{g} = T_{e}G$. The Lie Algebra ...
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0answers
35 views

Is it possible for distinct geodesics to be equivalent over a finite segment?

Is it possible for two geodesics $\gamma_1, \gamma_2$ to be identical within a finite interval without being identical outside the interval? IOW: $\gamma_1(t) = \gamma_2(t)$ for $t \in (A,B)$ but ...
0
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4answers
48 views

Why do we use only compatible charts in the Theory of Manifolds?

I couldn't find a duplicate, although I think is a very common question. Given two charts, ($U_{1},φ_{1}$), ($U_{2},φ_{2}$), on a n-dimensional topological manifold M, such that: $U_{1} \cap ...
4
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1answer
44 views

example of a subset of a smooth manifold admitting a unique smooth structure making the inclusion an immersion, which is not a weak embedding.

A subset $S$ of a smooth manifold $M$ is called a weakly embedded submanifold (at least in Lee) if it admits a smooth structure making the inclusion an immersion, and such that for any other smooth ...
2
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1answer
27 views

how to visualise orthonormal frame bundle?

how to visualise the orthonormal frame bundle? The orthonormal frame bundles $O(\Sigma)$ of $\Sigma$ is the set of pairs $(x,H)$, where x is a point of $\Sigma $ and H is an orthonormal frame of ...
2
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1answer
68 views

Restriction of ${\rm spin}^c$ structures

Suppose I have an oriented 4-manifold $X$ with boundary $\partial X$ an rational homology 3-sphere. If the restriction map $${\rm Spin}^c(X) \rightarrow {\rm Spin}^c(\partial X) $$ is surjective then ...
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0answers
26 views

Why Laplace-Beltrami operator is so popular for 3D shape analysis.?

Apart from providing orthogonal basis in form of eigen functions what is the reason that Laplace-Beltrami operator is so popular in shape and point cloud processing.
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1answer
28 views

Find the transition functions and show that $M$ is non-orientable.

Let $M$ be the collection of all affine lines on the plane $\mathbb{R}^2$. Introduce an atlas of two charts on $M$. The chart $U_1$ consists of all non-vertical lines, and the line $L : y = a_1x +b_1$ ...
2
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1answer
99 views

Show that the Mobius strip is non-orientable

The Mobius strip is the 2D manifold $M$ with the atlas of $n$ cubic charts $U_i$, $1 ≤ i ≤ n$, with coordinates $(x_i, y_i)$ satisfying $|x_i| < 1, |y_i| < 1$. Let $U_i^±$ be a part of $U_i$ ...
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0answers
28 views

Why is the canonical bundle 1 dimensional

The canonical bundle is defined to a bundle of $n$-form, so how can it be one dimensional?
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1answer
20 views

On the Existence of a Particular Local Coordinate System

Suppose $M$ is a topological manifold and $(U,\phi)$ a local chart around $p\in M$. Is it always possible to find a chart $(U,\psi)$ such that $\psi(U)=B$ where $B$ is, say, the unit ball in ...
12
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2answers
141 views

How do we construct an associated bundle $V_{n, q}(\omega)$ over $B$ with typical fiber $V_{n - q}(F)$?

Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q ...
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0answers
28 views

Show that a given set is a manifold with boundary

given $A:= \{ (x_1,x_2,x_3) \in \mathbb{R} : x_1^2 + x_2^2 + x_3^2 = 18, x_3 \leq 2\}$ show that $A$ is a manifold with boundary and calculate $ \delta A$ where $\delta A$ is the boundary of A. I ...
3
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1answer
37 views

Vector Bundles: Continuity of map between total space implies homeomorphism.

In Spivak's "A Comprehensive Introduction to Differential Geometry" Spivak defines a vector bundle as a tuple: $(E, \pi, B, \bigoplus, \bigodot),$ where $E$ is the total space, $B$ is the base space, ...
2
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0answers
43 views

Literature Request: Stochastic Differential Geometry

I've in my studies taken (introductory, at the masters level) courses on both stochastic calculus, differential geometry (both elementary at the level of Pressley's book, and more advanced at the ...
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0answers
26 views

Checking if a tangent bundle is trivial

I don't know how to check if tangent bundle $TP$ of the surface $P$ is trivial. Are there any general methods to deal with this problem? For example how to check it for $P\subset \Bbb{R}^3$ arisen by ...
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0answers
259 views

Solutions manual for Analysis On Manifolds

A few months ago,I wanted to learn something fundmental about manifolds. From highly recommend , I decided to choice Analysis on Manifolds by James R.Munkres as my self-learning textbook.Until now ,I ...
2
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3answers
43 views

Can I prescribe the geodesics?

Consider $J$ an open interval of $\mathbb{R}$. An inner product on $\mathbb{R}$ is necessarily of the form $(u,v) \in \mathbb{R}^{2} \, \mapsto \, auv$ with $a > 0$. Therefore, a Riemannian ...
0
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1answer
35 views

On the implicit function theorem in more dimensions.

In class we stated and proved the implicit function theorem in the case where we have an open set $A \subset R^2$ a function $f:A \rightarrow R, \ f \in C^1_A$ and a point $ (x_0, y_0) \in A$ s.t. ...
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0answers
35 views

$S^1 \times \mathbb{R}$ is diffeomorphic to $TS^1$

I know this question has been asked, but I've been given a different map that I am not entirely sure how to handle. I am fairly lost on this problem. Show that the map $$F: S^1 \times \mathbb{R} ...
1
vote
1answer
39 views

Lie Bracket Calculation for Integral Curves

I am trying to derive a Lie bracket, and then find the related integral curve at the point $(x_0,y_0)$. The problem gives the vector fields $X = y \frac{\partial }{\partial x}$ ,$Y = \frac{x^2}{2} ...
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0answers
36 views

Showing that Heisenberg group is a Lie Group.

We define the Heisenberg group $H^{n}$ for $n\geq1$ as follows. As an analytic manifold $H^{n}=\mathbb{R}^{2n+1}$. We denote elements in $H^{n}$ by $(t_{i},q_{i},p_{i})$ with $t_{i}\in\mathbb{R}$ and ...
0
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1answer
34 views

tensor product of a line bundle with $\bigwedge^k T^*M$

I am reading a source that says: Let $L \to M$ be complex line bundle. Define $\Omega^k(M,L)=C^\infty(M,L \otimes\bigwedge^k T^*M)$. Can someone explain thoroughly what this means? What does it mean ...
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5answers
532 views

A Compact Hausdorff Space with no Manifold Structure? [closed]

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
3
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0answers
48 views

A $k+1$ Manifold whose boundary is the solution set to the equation $f(\vec{x})=\vec{0}$

Let $f: \mathbb{R}^{n+k} \to \mathbb{R}^n$ be of class $C^r$. Let $f_1, ..., f_n$ be components of $f$. Define $$M=\{\vec{x} | f(\vec{x})=\vec{0}\},$$ $$N=\{\vec{x} | f_1(\vec{x})=0, ..., ...
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votes
0answers
26 views

Detailed proof (submersion) : show that the differential is surjective

I'm currently studying manifolds and wanted to have a detailed insight on a part of some proof. This might be very easy, but I can't find the good words to express the correct idea. My definition of ...
2
votes
1answer
28 views

Use of implicit function theorem in showing that $f(x) \leq a$ is a submanifold with boundary

This question comes from a statement in John Milnor's "Morse Theory" on page 4. Let $f: M \to \mathbb{R}$ be a smooth function on a manifold $M$. Milnor claims that if $a$ is not a critical value of ...
3
votes
1answer
78 views

Property of second Steifel-Whitney class?

Let $M$ be manifold, $n = 4$. Is $w_2$ special in in the regard it's the only thing of $H^2(M, \mathbb{Z}_2)$ where $w_2 \cup \tau = \tau \cup \tau$, $\tau \in H^2(M, \mathbb{Z}_2)$ or not? I wondered ...
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0answers
24 views

Compute the degree of $\phi:(x_1,x_2,x_3,x_4) \mapsto (x_1,-x_3,-x_2,x_4)$. Does $\phi$ preserve orientation?

Consider the map $\phi:S^3 \rightarrow S^3$ given by $\phi(x_1,x_2,x_3,x_4)=(x_1,-x_3,-x_2,x_4)$. Compute the degree of $\phi$. Does $\phi$ preserve orientation? First I want to point ...
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0answers
35 views

Lifting of a Diffeomorphism to an Orientable Double Cover

Let $M$ be a non-orientable smooth $d$-dimensional manifold. Let $\tilde{M}$ be the oriented double cover for $M$; and let $f\in Diff^{1+\beta}(M)$. Define $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ ...
2
votes
1answer
44 views

When is a manifold also a vector space?

My question arises from this definition: Poincare group is the group of Minkowski space-time isometries. Which means that it leaves the space-time intervals unchanged. Now here is my understanding: ...
2
votes
1answer
64 views

resources for classical gauge theory

As a prospective grad student, I would like to get an entry level introduction to classical (i.e. non-quantum) gauge theory. Please direct me to resources suitable for a novice.
3
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1answer
46 views

How to find the tangent space of a general submanifold?

Given a submanifold $(S,\phi)$ of a manifold $M$, how do we find the subspace of $T_pM$ that is equal to $T_pS$ for $p\in \phi(S)$. I know how to do it for level sets. Is there a way for general ...
2
votes
0answers
26 views

Writing $\mathrm{SO}(2)$ as the zero-set of a function

Here I'm assuming $M_{2 \times 2}(\mathbb{R}) \cong \mathbb{R}^{4}$. The definition of $\mathrm{SO}(2)$ is: $\mathrm{SO}(2)=\{ \ A \in M_{2 \times 2}(\mathbb{R}) \ | \ \det(A)=1 \mathrm{\ and\ ...
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0answers
86 views

Einstein notation

I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds). I want to write the following thing: Let $X$ be a smooth manifold. Choosing local ...
6
votes
1answer
69 views

Is $[N]^\#([N])$ congruent to $w_n(\nu_N)([N])$ mod $2$, where $\nu_N$ is the normal bundle of the embedding of $N$ in $M$?

Let $M$ be a closed, smooth, orientable $2n$-manifold, and let $N$ be a closed, smooth, orientable $n$-submanifold. Let $[N]^\#$ denote the cohomology class (Poincaré) dual to the homology class ...
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0answers
37 views

Proof of relation between normal of a surface and principle curvatures of surface.

If F(x,y,z) is a scalar function. Then how to prove that, $$\nabla . n = K_1 +K_2$$ where n is normal to surface of constant $F$ given as $$n=\frac{\nabla F}{|\nabla F|}$$ $K_1$ and $K_2$ are ...
0
votes
1answer
72 views

Proof that the special linear group $\mathrm{SL}(n,\mathbb{R})$ is a smooth manifold

This is my definition of a smooth manifold I am supposed to work with: Let $\mathcal{M} \subseteq \mathbb{R}^{n}$. The set $\mathcal{M}$ is a $k$-dimensional smooth submanifold of $\mathbb{R}^{n}$ ...
0
votes
1answer
24 views

How can I calculate this integral of a differential form in a surface?

I'm trying to integrate the 2-form $\omega = A(y) dx \wedge dy - dx \wedge dz + B(z)dz \wedge dy $ in the set $R_f=\{(x,y,z),\quad z=f(x,y)\quad x^2+y^2 \neq 1 \}$ with $f$ a differentiable function ...
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1answer
28 views

Proving $\mathrm{GL}(n,\mathbb{R})$ is a smooth manifold

Consider the set $\mathrm{GL}(n,\mathbb{R}) = \{ \ A \in M_{n \times n}(\mathbb{R}) \ | \ \mathrm{det}(A) \neq 0 \ \}$. I'm trying to show that this is smooth submanifold of $\mathbb{R}^{n^{2}} \cong ...
2
votes
1answer
38 views

Explicit procedure for integrating densities (twisted $n$-forms)?

I'm aware of several slightly different (yet equivalent) definitions for the density bundle of a non-orientable manifold. Unfortunately, I've been unable to make sense of integration in any of them. ...
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0answers
49 views

Can the concept of orientability be applied to more general spaces?

Today a friend asked me if the Moebius strip with one segment identified to a point is orientable or not. The first thing I replied is that it is not a manifold so you can't define orientability in ...
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0answers
16 views

How can I understand that this mapping preserves the orientation of the boundary of this manifold?

Let $M$ be the cylinder of radius 1 (with $z$ between 1 and -1) and $f: M \to M$ the application defined as $f(\cos(\theta), \sin(\theta), t) = (\cos(4\theta), \sin(4\theta), -t)$. I want to give an ...
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1answer
58 views

How can I calculate the integral $\int_M F^* \omega$?

I got stuck in the following problem. Let $M$ be the manifold defined by the equation $x^2+y^2+z^4=1$ and $F: M \to S^2$ defined as $F(x,y,z)=(x,y,z^2)$. I have to calculate the integral $\int_M F^* ...
2
votes
1answer
26 views

Expression of a given vector field for the stereographic projection of the sphere

I have got stuck trying to solve the following problem. Let $X=-zx \frac{\partial}{\partial x} -zy \frac{\partial}{\partial y} + (1-z^2) \frac{\partial}{\partial z}$ be a vector field in ...
0
votes
2answers
31 views

Does the forgetful functor from smooth manifolds to sets preserve colimits?

It is easily shown that the forgetful functor $F: \mathbf{Man} \to \mathbf{Set}$ preserves limits ($F$ is representable), but does it preserve colimits? It certainly preserves all examples of ...
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1answer
51 views

Calculating Volume of surface of unit sphere

I am trying to understand the proof for $w_n = 2\pi^{n/2}/\Gamma(n/2)$ where $w_n$ is the volume of the surface $S_n$ of the n-dimensional unit sphere $K_n$. There is stated that $Vol(K_n) = ...
0
votes
0answers
18 views

Grassmannian Non-Convex

The Grassmannian manifold $Gr(r,V)$ defines the set of $r$-dimensional linear subspaces of the vector space $V$. My question is, in general, what is the simplest way to see that $Gr(r,V)$ is a ...
1
vote
2answers
73 views

Rank of Jacobian Matrix for the Stereographic Projection

With the definition $S^{n} = \{\ \mathbf{x} \in \mathbb{R}^{n+1}\ | \ ||\mathbf{x}|| = 1\ \}$, and the function $\ f:\mathbb{R}^{n} \to S^{n} \setminus \{ (0,...,0,1) \}$ defined by: $f(\mathbf{u}) ...