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

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1answer
185 views

If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold?

I am reading some basic context books about topology (i.e. The Poincaré Conjecture, by Donal O'Shea between others) and following this open Topology and Geometry video lectures of the brilliant ...
1
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1answer
46 views

Multivariable Calculus: Manifolds

Problem Let $M$ be the set of all points $(x,y) \in \mathbb{R}^2$ satisfying the equation $$xy^3 + \frac{x^4}{4} + \frac{y^4}{4} = 1 $$ Prove that $M$ is a manifold. What is the dimension of $M$? ...
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2answers
24 views

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

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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2answers
80 views

Is there a unique preferred connection on a general manifold?

I was wondering whether, for a given finite-dimensional manifold, the connection $\nabla$ exists and is uniquely defined? Afais for Riemannian manifolds, there exists always exactly one Levi-Civita ...
4
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0answers
40 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
42 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 ...
4
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1answer
112 views

Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
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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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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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0answers
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 $\mathbb{R}^{d}$ ...
2
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1answer
173 views

Spivak Calculus on Manifolds, Theorem 5-2

In the proof Theorem 5-2 of Spivak Calculus on Mannifolds how is \begin{align*} V_2\cap M=\{f(a):(a,0)\in V_1\}? \end{align*} (That $\{f(a):(a,0)\in V_1\}=\{g(a,0):(a,0)\in V_1\}$ is clear.) Edit: ...
2
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0answers
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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0answers
12 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, ...
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 ...
15
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1answer
388 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
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1answer
55 views

Understanding tangent space basis

Consider our manifold to be $\mathbb{R}^n$ with the Euclidean metric. In several texts that I've been reading, $\{\partial/\partial x_i\}$ evaluated at $p\in U \subset \mathbb{R}^n$ is given as the ...
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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 ...
3
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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 ...
2
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1answer
26 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 ...
0
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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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5answers
1k views

Why the surface of the sphere is not a Euclidean space?

In wiki, I see a description in manifold: "Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, but in ...
2
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1answer
42 views

Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
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1answer
40 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 ...
0
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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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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 $\mathfrak{g}$) ...
0
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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 ...
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0answers
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 ...
0
votes
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 ...
0
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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 ...
6
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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
48 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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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 ...
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 ...
1
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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?
2
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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, ...
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 ...
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 ...
5
votes
5answers
3k views

Simpler definition of manifold

I'm new to topology but I must understand how it works to progress in my research. First, can anybody point me to a document that introduces topology in a "gentle" way. What pre-requisites do I need ...
8
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1answer
84 views

What are the 8 non-compact Euclidean 3-manifolds?

I have found several sources that mention that there are eight non-compact Euclidean 3-manifolds, with four orientable and four non-orientable. There are two standard references for this, but ...
0
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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 ...
0
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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 ...
4
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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
17 views

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 ...
0
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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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0answers
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 ...
0
votes
1answer
31 views

Continuity in definition of Induced Functional Structure

I have a really simple question, however I am confused. Bredon's Topology and Geometry gives definition of Induced Functional Structure as follows: Suppose $F_x$ is a functional structure on space ...
7
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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 ...