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

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Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
4
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36 views

Higher-Order Differential Operators as Vector Fields

On a $C^\infty$ manifold $M$, one can produce the tangent space $T_p M$ at a point by equivalence classes of tangents to smooth curves through the point $p$. When realised this way, the tangent ...
3
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1answer
41 views

Involutive distributions?

How do we check exactly that a distribution is involutive? I have the following definition in my book: A $k-$dimensional distribution $\Delta$ on a manifold $M$ is a smooth choice of a k-dimensional ...
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0answers
22 views

Prove winding number is the same as index of a vector field. [on hold]

I'm trying to prove that the winding number and the index around a point in a vector field are the same. I know that the index is sometimes defined as the winding number but I'm working with the ...
5
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54 views

Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
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11 views

Optimization on SE(3) with matrix logarithm

I am trying to optimize the following equation on manifold SE(3). $$Z(e^{\epsilon}) = \text{logm}{((e^{\epsilon}X)^{-1}W^{-1}e^{\epsilon}XY)}$$ Note that $W, X, Y, e^{\epsilon} \in SE(3)$ and $W, X, ...
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1answer
43 views

Non injective continuous maps

Motivated by comments on this question we ask the following question: Let $f:M\to M$ be a continuous map where $M$ is a compact manifold and $f$ is not injective. Are there necessarily ...
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1answer
49 views

topic between algebra and geometry [closed]

I have to do an exam on Differential Geometry and my teacher wants that I prepare a choosen topic, outside lectures program, that I will talk about at the oral part of the exam. I am interested in ...
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1answer
25 views

The restriction fo covering to a component is a covering map onto its image.

I am reading Lee's Introduction to Topological Manifolds. I got stuck on the problem 11-7 on pages 303. The below is the problem. Prove : If $q: E \rightarrow X$ is a covering map and $A \subseteq ...
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0answers
11 views

Transformation of a subset of compact Jordan sets to manifolds

Let $T$(for e.g. $[0,1]^2$) be a Jordan compact sets and $\tau$ be a "smooth enough one-to-one" transformation, i.e.($\tau: [0,1]^2 \rightarrow [0,1]^2 $). Lets take a subset of Lebesgue measure ...
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1answer
21 views

p-simplex spanned by elements in the boundary of the unit ball of Thurston norm is in the the boundary of this unit ball.

in completing my thesis I have reached a momentary impass. I am trying to solve an exercise given in the book "Foliations II" by Candel and Conlon. In particular, Exercise 10.4.1, and I can't seem to ...
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61 views

Understanding the “shape” of a singular Riemann surface

Consider the singular Riemann surface given by the following expression: $$z^d w^d-z^d-w^d+t=0\ ,$$ where $t$ is a parameter in $(-1,1)$ and $d$ is a positive integer greater than 2. For $t\neq0$ the ...
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1answer
21 views

real coordinates of a complex manifold

I have a naive question about real coordinates of a complex manifold. Let's consider 1-dimensional case for simplicity. Let $X$ be a Riemann surface and $z$ be a local complex coordinate. Then one ...
2
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1answer
74 views

How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
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1answer
39 views

Basic examples topological manifolds with boundary

I've just started to study differential geometry and I've some problems with the first definitions. We have defined a topological manifold with boundary of dimension n as a topological space $M$ such ...
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0answers
35 views

n-dimensional lattice as a collection of lower dimensional spaces.

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
2
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3answers
46 views

Group actions on manifolds - exponential map

Let $M$ be a smooth manifold. Suppose $K$ is a Lie group (with Lie algebra $\mathfrak{k}$) acting EDIT: TRANSITIVELY on $M$ from the left and $G$ is a Lie group (with Lie algebra $\mathfrak{g}$) ...
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13 views

Transitive group actions on Principal bundles

I have a question in regards to page 107 of Kobayashi & Nomizu's Foundations of Differential Geometry Setup: Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$ whose Lie ...
3
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0answers
16 views

Heegaard splitting via a Morse function - twisted union or not?

Let $M$ be a smooth, closed, connected, oriented 3-manifold and let $f: M \rightarrow \mathbb{R}$ be a self-indexing Morse function. Since $\frac{3}{2}$ is a regular value of $f$ it follows from Morse ...
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1answer
14 views

Critical and regular values of height functions on a closed hypersurface

Let $M$ be a closed connected hypersurface of $n$-dimensional in $\mathbb{R}^{n+1}=\{(x^1,\cdots,x^{n+1})\}$ and let $\nu$ be a smooth unit normal vector field of $M$ at $\mathbb{R}^{n+1}$, $H$ be the ...
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0answers
34 views

equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
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1answer
82 views

Tangent space manifold

Let M be a differentiable manifold of dimension m and also let $\{\xi_1,\dots,\xi_m\}\subset \text{T}_pM$ be an linearly independent set of the tangent bundle of M at a certain point $p\in M$. I have ...
3
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2answers
47 views

Push-forward of vector fields by local isometries

I am studying Riemannian Manifolds by John Lee, and there is this lemma: Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if ...
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1answer
57 views

What's the geometrical meaning of immersion?

Does that just mean to different tangent vectors, their images are different tangent vectors?
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2answers
53 views

Does a proper map have to be continuous?

In Pollack's differential topology, the proper map is defined by the preimage of every compact set is compact. Here it doesn't require the map to be continuous. However, in his following claim, to a ...
3
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1answer
83 views

Looking for a good alternative to 'An introduction to manifolds' by Loring W. Tu

I'm currently studying some basic theory about manifolds from the book 'An introduction to manifolds' by Loring W. Tu. The problem I have with this book is that there are very little exercises, and ...
2
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1answer
56 views

Area form and surface area

I know how one can define the surface area via the charts of a surface in $\mathbb{R}^3.$ click here for instance Now, I read that the canonical surface area form for such a surface with surface ...
2
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1answer
52 views

Volume of Manifold with zero Lebesgue measure

Let $M$ be a smooth manifold in $\mathbb{R}^n$. If Lebesgue measure of $M$ is zero i.e $l(M)=0$, does it mean that volume of manifold is also zero i.e $Vol(M)=0$? Are they the same thing (volume and ...
4
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1answer
39 views

Local isometries preserve geodesics?

Question: It is well known that if $\varphi:M\to \tilde{M}$ is an isometry between Riemannian manifolds, then $\varphi$ maps geodesics of $M$ to geodesics of $\tilde{M}$. I am wondering if it is ...
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0answers
32 views

Isotopy: Definition

An isotopy is a homotopy from one embedding of a manifold $M$ in $N$ to another such that at every time, it is an embedding. In this definition, I am wondering why $M$ and $N$ are required to be ...
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Is every compact manifold with boundary a manifold of bounded geometry?

If $M$ is a compact Riemann manifold with boundary, does it have bounded geometry, which means that the injectivity radius of the manifold is positive and every covariant derivative of the Riemannian ...
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2answers
53 views

Action of the fundamental group

Suppose that $M$ is a smooth manifold. Is it true that the fundamental group $\pi_1(M)$ always acts on $M$? If so, how this action is defined? EDIT: Of course I want my action to be nontrivial, say ...
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2answers
26 views

Are $X=M \times [0,T]$ and $\partial X$ smooth compact manifolds when $M$ is smooth compact Riemannian manifold?

Let $X=M \times [0,T]$, where $M$ is a smooth and closed compact Riemannian manifold. I want to know if: $X$ is smooth compact manifold, and if $\partial X$ is smooth compact manifold? I am not ...
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69 views

SO(n) orientable?

I have to answer the question whether $SO(n)$ is orientable or not...Actually I have no idea - could someone help me? I already know that $SO(n)$ is a $n(n-1)/2$-dimensional manifold, but how can I ...
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1answer
135 views

Smooth map $S^1 \to S^2$ can not be surjective

Why cannot a smooth (or piecewise linear) map $S^1 \to S^2$ be surjective? There are space-filling curves, but the usual examples have very "twisty" definitions. UPD A bit of background for this ...
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29 views

Why do we require differential manifolds to be Hausdorff? [duplicate]

Among the requirements for a differential manifold $M$ is that it be connected and Hausdorff. What fails if a manifold is not Hausdorff?
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1answer
27 views

Transitive Lie group actions and surjectivity of maps

I am reading a paper at the moment and I have come across two statements which I want to understand. Here is the setup: Suppose that $G$ is a Lie group which acts on a manifold $E$ differentiably and ...
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1answer
81 views

Confused (disoriented?) by questions about orientation

I believe I have a reasonable basic understanding of orientation. Yet, i'm finding myself utterly confused when facing a specific question. Here are several examples: Exhibit an ordered basis ...
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0answers
15 views

Determining if an equation represents (?) a Riemann surface

This is my first exposure to Riemann surfaces. I have studied complex analysis in an introductory course, and spent the last few weeks learning a little bit of deeper theory with Conway's Functions of ...
2
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2answers
44 views

What is the Euclidean topology on $\mathbb{R}^0$ like?

I am trying to prove that a topological space $(X,\mathscr{T})$ is a $0$-manifold if and only if it is a countable discrete space. In the process I have to show that there exist a homeomorphism from a ...
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6 views

Maximally symmetric manifold with boundary and non-vanishing extrinsic curvature?

I was wondering if the following requirements are compatible: Given a $d$-dimensional manifold with boundary $M$ with $\partial M\neq \emptyset$ endowed with a metric $g$. The following conditions ...
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3answers
92 views

Homeomorphic manifolds have the same dimension

So I want to prove: If two manifolds $M$ and $N$ are homeomorphic then $dim(M) = m = n = dim(N)$. My idea was to use the property of the manifolds that they are locally homeomorphic to the ...
3
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1answer
47 views

Hermitian metric on $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space

Consider the line bundle $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space. Locally is descriebd by $\{U_a,g_{ab}\}$ where $U_a=\{z_a\neq0\}$ is the standard covering of the projective ...
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1answer
62 views

What is the dimension of the space of planes in $\Bbb R^3$?

What is the dimension of the space of planes in $\Bbb R^3$ and how do we reach the answer? Clarification: What I am searching for is what is the least number of parameters that I need. For example, ...
2
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0answers
20 views

Tubular neighborhood by restricting the Riemannian exponential map

Let $M$ be a Riemannian manifold (possibly non-compact, possibly non-complete) and $N\subseteq M$ a smooth submanifold (possibly non-compact). Does there exist a continuous $\mu\colon M\rightarrow ...
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1answer
44 views

Several statements about $\mathbb{R}$ with chart defined by $f(x)=x^3$

I think I managed to show this statements but I am not sure about it. Since this is common problem in differentiable manifolds I was wondering if anybody has (or may write) a solution. Let $X$ be a ...
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1answer
42 views

A closed set is not a submanifold

Can someone explain me why the set $A:=\{(x,y)\in \mathbb R^2: x\geq 0\}$ is not a submanifold? I also got another (easy) question: In our lecture we are always talking about submanifolds. We ...
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40 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
3
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1answer
48 views

group cohomology of 3-manifolds

If $M$ is a closed aspherical 3-manifold with first fundamental group $G$, then cohomology groups of $G$ and $M$ are isomorphic because $M$ is a Eilenberg-MacLane space $K(G,1)$. In particular there ...
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82 views

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...