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

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7
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1answer
52 views

Reference on manifolds with corners

Is there a systematic treatment of (finite dimensional) manifolds with corners in the literature which carefully introduces all usual differential topological notions (submanifolds, embeddings, etc.) ...
3
votes
0answers
36 views

Riemannian surface, identity relating scalar curvatures and Laplacian [closed]

Let $S$ be a Riemannian surface, i.e. a $2$-dimensional manifold, with metric $g$. Define a new metric $\tilde{g}$ by $\tilde{g} = e^fg$ for some smooth function $f$. If $s_{\tilde{g}}$ and $s_g$ are ...
2
votes
1answer
71 views

Is a Kähler manifold necessarily symplectic?

Let $M$ be a Riemannian manifold. If we pick a basepoint $p \in M$, then for any smooth path $\gamma: [0, 1] \to M$, parallel transport along $\gamma$ induces an automorphism $g_\gamma \in ...
0
votes
1answer
31 views

How do I prove that a surface defined by two functions is a manifold?

I am trying to show that $$\{(x, y, z)\mid z=x^2+y^2, z\leq 2\}$$ is a manifold. I am trying to express it as a set where $f(x, y, z)\geq 0$ for some smooth $f$ on an open set, but as the set ...
3
votes
1answer
62 views

Riemannian manifold, $\alpha \in \Omega^p(M)$ parallel implies $\alpha$ is closed?

Let $M$ be a Riemannian manifold, and let $\alpha \in \Omega^p(M)$ be parallel; i.e. suppose $\nabla \alpha = 0$ where $\nabla$ is the Levi-Civita connection. Does it necessarily follow that $\alpha$ ...
1
vote
1answer
31 views

Intuition of section of a hermitian line bundle

Can someone explain to me intuitively and without much technical stuff the following: A hermitian line bundle is a complex line bundle with a hermitian metric. I think of this as a bundle over my ...
2
votes
1answer
30 views

Compact 3-manifold implies finite triangulability

I know that it's a theorem by Moise that every compact 3-manifold admits a finite triangulation but to me the astounding part of that statement is the existence part instead of the finite one. So I ...
1
vote
2answers
47 views

Smooth functions between manifolds and subsets of manifolds

I'll be quoting from the Wikipedia page on smoothness. Smooth function between manifolds are defined as follows: If $F$ is a map from an $m$-manifold $M$ to an $n$-manifold $N$, then $F$ is smooth ...
3
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0answers
37 views

Primary obstruction to the existence of a cross-section of $V_{n - q}(\omega)$ is a cohomology class in $H^{2q+2}(B, \pi_{2q+1} V_{n - q}(F))$?

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 + ...
0
votes
1answer
16 views

Continuity of the Length of shortest arc containing angles

Here's an interesting question! Let $x=(x_1,x_2,\dots,x_n)$ points on the unit circle of $\mathbb{R}^2$. And let $V_{\min}(x)$ be the length of the shortest arc containing all the points ...
0
votes
1answer
34 views

How do I prove that a set of equations define a submanifold?

I have two equations of 4 variables, and want to prove that they define a submanifold. I showed that two of the variables are defined by the other two, and wanted to get a graph out of it to show ...
1
vote
2answers
49 views

Understanding of tangent space

I have some understanding issues with the following theorem. Let $M \subset \mathbb{R}^N$ be a d-dimensional smooth manifold. $p \in M$ with $(\phi, U)$ a chart around $p$. Then $T_pM = (d\phi ...
0
votes
1answer
35 views

Smooth function on a manifold not dependent on coordinate chart

I'm having trouble with the proof of the following remark from page 59 of Tu's book on Manifolds. The part I'm worried about is where he gets that $\phi\circ \psi^{-1}$ is $C^\infty$. Is he allowed to ...
0
votes
0answers
17 views

Proving that the following is a submanifold with boundary

Let $\mathbb{H}^{k} = \{ \ \mathbb{x} = (x_{1}, ..., x_{k}) \in \mathbb{R}^{k}\ | \ x_{k} \geq 0\ \}$. Definition: If the following holds, then $\mathcal{M}$ is a $k$-submanifold of $\mathbb{R}^{n}$ ...
1
vote
1answer
31 views

Mapping the Upper Half plane to the Unit Ball

Consider the sets $\mathbb{H} = \{ \ (x_{1}, ..., x_{n}) \in \mathbb{R}^{n} \ | \ x_{n} \geq 0\ \}$ and $B_{1}(\mathbf{0}) = \{ \ \mathbf{x} \in \mathbb{R}^{n}\ | \ ||\mathbf{x}|| \leq 1 \ \}$. Does ...
1
vote
1answer
31 views

Examples of quotient manifolds which are not locally trivial fibrations?

Let $X$, $Y$ be differentiable manifolds, and $f : X \to Y$ a smooth surjection. Then $Y$ is said to be a quotient of $X$ if 1) $Y$ has the quotient topology 2) A function $g : Y \to \mathbb{R}$ is ...
0
votes
2answers
22 views

Proving non-compactness of a manifold

I have been trying to solve the following problem: Let $M \subset \mathbb R^3$ be the set of points $(x,y,z) \in \mathbb R^3$ at which $xy + xz + yz = 1.$ Prove that $M$ is a $2$-dimensional manifold. ...
0
votes
0answers
15 views

Does a derivative with respect to a function and a derivative with respect to variable commute?

Lets say $\phi:M\to\mathbb{R}$ is a function defined on the manifold $M$ and $\{x^\mu\}$ are the coordinates of some chart $U$ on it. I am trying to check if $ ...
1
vote
1answer
33 views

Is the curve $S=\{(|\sin(t), \cos(t) \exp(t)) \in\mathbb{R}^2 \mid t \in(0,{3\pi\over4})\}$ a manifold in $\mathbb{R}^2$?

Is the curve $S=\{(|\sin(t), \cos(t) \exp(t)) \in\mathbb{R}^2 \mid t \in(0,{3\pi\over4})\}$ a manifold in $\mathbb{R}^2$? I feel it is a manifold, as when you look at the graph, it does not ...
3
votes
1answer
45 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 ...
1
vote
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 ...
1
vote
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
votes
4answers
50 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
votes
1answer
48 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
votes
1answer
29 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
votes
1answer
69 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 ...
0
votes
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.
0
votes
1answer
30 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
votes
1answer
104 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 $U_i$ ...
1
vote
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?
0
votes
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
votes
2answers
142 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 ...
1
vote
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
votes
1answer
41 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
votes
0answers
47 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 ...
0
votes
0answers
27 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 ...
1
vote
0answers
332 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
votes
3answers
44 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
votes
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. ...
0
votes
0answers
36 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} ...
0
votes
0answers
41 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
votes
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 ...
5
votes
5answers
535 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
votes
0answers
50 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, ..., ...
0
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
31 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
88 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 ...
2
votes
0answers
34 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 ...
0
votes
0answers
36 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 $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ ...