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Questions tagged [manifolds]

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

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1answer
15 views

integral curve starting at a zero of a vector field

This is a question from Loring Tu's book "Introduction to manifolds" (Page-161 14.6(b)) Show that if X is the zero vector field on a manifold M, and ct(p) is the maximal integral curve of X starting ...
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14 views

Question on the dimension of the boundary of a bounded domain

Motivated by this post Dimension of boundary of a bounded domain; what to use for Sobolev inequalities and since the provided answer didn't help me that much, I thought I should just ask once more. ...
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2answers
26 views

Is $W$ a finite-dimensional vector space in Proposition 3.13?

Is $W$ a finite-dimensional vector space in Proposition $3.13$?
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28 views

Proof: A tangent space of the manifold of SPD matrices is the set of symmetric matrices

The set of SPD matrices, $\mathbb{P}_n := \{X \in \mathbb{R}^{n \times n} | X=X^T, X \succ 0 \} $, forms a differentiable manifold. Claim: The tangent space at a point, $A, T_A\mathcal{P}_n$ is the ...
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1answer
13 views

Showing that a locus is a sub-manifold

I'm self-studying differential geometry using Frankel's ``The Geometry of Physics". The first problem (1.1(1)) is about determining whether or not the locus $$x^2+y^2-z^2 = c $$ is a submanifold in $\...
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0answers
21 views

Why does the exponential map have this form?

I believe I understand what the exponential map does, in informal terms. However I cannot relate this to the equation I see before me in the papers. The exponential mapping function for a symmetric ...
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28 views

Express the volume of $S^n(a)$ in terms of the volume of $B^{n-1}(a)$

I'm working on an exercise which appears in a chapter about integration on manifolds. It asks to express the volume of $S^n(a)$ in terms of the volume of $B^{n-1}(a)$. Here $S^n(a)=\{x\in\mathbb{R}^{n+...
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0answers
25 views

When are homotopy-equivalent 4-manifolds s-cobordant?

Suppose $X$ and $Y$ are closed 4-manifolds, not necessarily simply connected. Such manifolds are said to be s-cobordant if there is a 5-manifold $W$ with $\partial W = X \sqcup Y$ such that the ...
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0answers
56 views

Does the integral cohomology ring determine the ring structure with other coefficients?

Let $A$ be an abelian group. By the universal coefficient theorem, the cohomology group of a manifold with coefficient $A$ is determined by the integral cohomology group. How about the ring structure? ...
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1answer
38 views

Why are map projections of the Earth not charts?

The Earth is a classic example of a 2D manifold. Looks Euclidean to us, but is most definitely curved. I am self teaching some differential geometry and I don't quite understand the difference ...
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13 views

Are the two definitions of that a map $F:A\to N$ is smooth on $A$ equivalent?

In the above pictures, we can see two definitions of that a map $F:A\to N$ is smooth on $A$, are they equivalent?
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1answer
32 views

What is the difference between a chart and a tangent space?

To my lay-person mind, a chart is a one-to-one function that maps an area on a manifold to a euclidean space of equal dimension. Then I understand a tangent space to be the space of vectors that are ...
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1answer
85 views

How does a differential form looks on a matrix manifold?

I want to know how does a differential form looks in a matrix manifold. For example, given that the special linear group $$SL(n,R)$$ of all matrices with determinant 1 is a manifold, how looks a 1-...
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1answer
38 views

Interpretation of the Lie algebra of a Matrix Lie group

I'm looking for an intuitive explanation of the meaning of the Lie algebra for a matrix Lie group from a differential geometry perspective. Right now, the procedure I've been following is using the ...
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36 views

Is possible to figure out a kind of $negative$ immersion as a *truth function* of 'positive' immersion? [on hold]

ATTENTION: This is not a question, but a reflection on a possible meaning of the term 'negative' in a context in which I want to define the notion of 'negative immersion' an immersion is a ...
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1answer
36 views

Euler characteristic of matrix manifolds

I'm reading through examples of computing Euler characteristic of manifolds. I know how to compute it for generic manifolds like sphere and torus. But what about matrix manifolds? I'd like to know how ...
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2answers
24 views

Why is the set of tangent vectors at 0 in R^m bijective with R^m itself?

Barden & Thomas write in An Introduction to Differentiable Manifolds that the set of tangent vectors at 0 in R^m is bijective with R^m? Why is this the case? Can 0 be replaced by an arbitrary ...
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1answer
147 views

Boundary of the set of points away from manifold is a hypersurface

(This is part of Problem 6-6 from Lee's Introduction to Smooth Manifolds textbook.) Suppose $M\subset \mathbb{R}^n$ is a compact embedded submanifold. For any $\epsilon>0$, let $M_\epsilon$ be ...
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Generalising dual triangulation of manifolds

We know the following “geometric” version of Poincaré duality: Let $M$ be a closed $m$-dimensional manifold and let $\mathfrak{X}_*$ be a finite simplicial complex with $|\mathfrak{X}_*|=M$. We can ...
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1answer
109 views

Existence of transverse homotopy between knots in a 3-manifold

I have a 3-manifold $\Sigma$ and two homotopic embedded knots $K_{0}(t): S^{1} \to \Sigma$ and $K_{1}(t): S^{1} \to \Sigma$. I wish to refine the homotopy between them to a "transverse homotopy" i.e, ...
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1answer
12 views

Euclidean Connection and Constant Vector fields

Let $\nabla$ be the Euclidean connection on a manifold $M = \mathbb{R}^n$. The definition I'm following is if $X, Y$ is are smooth vector fields on $M$ with $Y$ given by: $$ Y = \sum_{i}^{n} Y^i \frac{...
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1answer
33 views

How to find the integral submanifold? [duplicate]

Suppose $U \subseteq R^3$ is the subset that all three coordinates are positive. Let $D$ be the distribution on $U$ spanned by two vector fields: $X = y\frac{\partial }{\partial z}-z\frac{\partial }...
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0answers
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Find the tangent space of Ellipsoid $M = \{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}$

Find the tangent space of $$M = \{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}$$ So I know the formula of tangent space for a manifold represnted by $F$ such that $F=0$: it is $ker (...
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0answers
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Quotient group as a manifold

Let $V_k(n) \subset \prod _{i=1} ^k S^n$ the real Stiefel space endowed with subspace topology and defined via $$V_k(n) := \{(v_1, v_2, ..., v_k) \vert \text{ } v_i \bot v_j \text{ for } i \neq j \...
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0answers
13 views

Why is this implying a normal vector field?

Suppose $\omega$ is a $n-1$ form on a $n-1$ dimensional manifold and $(a_1(x)dx_1 + ... + a_n(x)dx_n )\wedge \omega = c\Omega $, with $c \neq 0$ and $\Omega =dx_1\wedge...\wedge dx_n$. Moreover $(a_1(...
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2answers
39 views

Show there can be no co-ordinate patch at this point

I am attempting to prove that the subset of $\mathbf{R}^3$ satisfying $x^2 + z^2 = y^2$ is not a surface, where a surface is a subset of $\mathbf{R}^3$ for every point in which there is a co-...
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0answers
42 views

Why is the wedge product non zero?

Suppose $\omega$ is a non vanishing $n-1$ form on a $n-1$ dimensional submanifold $M \subset \mathbb{R}^n$. Why is $(a_1dx_1+...+ a_ndx_n)\wedge \omega$ nonzero if we know that there is at least one ...
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0answers
14 views

Question about proof of n-1 form inducing normal unit vector field

Suppose we have a $n-1$ dimensional manifold $M \subset \mathbb{R}^n$ and a non-vanishing $n-1$ form $\omega$ on $M$. This implies the existence of a normal unit vector field on $M$. The proof of ...
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coordinate systems produce submanifolds

In his book "Semiriemannian geometry with applications to relativity", Barrett O'Neil says on page 16 under definition 26 that "coordinate systems produce submanifolds.If T:U->R^n is a coordinate ...
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1answer
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“Seeing yourself” in a 3-manifold

In his book "The Shape of Space" (2nd ed), Jeffrey Weeks talks about a technique called "Cosmic Cristallography" to find the global topology of the space we live in. Starting with the assumption that ...
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1answer
46 views

Bijections between Manifolds of the same dimension

Given a continuous bijection between manifolds of the same dimension, does it have to be a homeomorphism? I know that this has a straightforward proof for compact Hausdorff space, be they manifolds ...
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1answer
37 views

n-1 form inducing normal unit vector field

Suppose we have a $n-1$ dimensional manifold $M \subset \mathbb{R}^n$ and a non-vanishing $n-1$ form $\omega$ on $M$. How would this imply the existence of a normal unit vector field on $M$?
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1answer
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Connected Sum Surgery

Is there any relationship between the connected sum operation and surgery theory? Is it possible to use surgery theory to "sew" two manifolds together and if so how is doing it by that approach ...
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0answers
48 views

Do any topology textbooks start from the closed set definition?

I am very interested in closed-set topologies, in particular because the induced topology on a manifold can be defined in terms of the smooth-scalar-field structure on that manifold as “a set of ...
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Submanifold of codimension 1 orientable iff there exists unit normal vector field.

Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Where a unit normal vector field is a section $\nu$ of the normal bundle $ TM^{\bot} \to M$. So the fibers are all the ...
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1answer
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Understanding why a finite dimensional vector space is a topological manifold

I wish to understand the details of why a finite dimensional vector space is a topological manifold, particularly following Jonh Lee's Introduction to smooth manifolds. I know that this questions has ...
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0answers
25 views

Need hint: Gradient of function on boundary of a connected subset of $\Bbb R^3$

Consider a scalar function $\Bbb R^3\to\Bbb R$ defined on a connected subset of $\Bbb R^3$. I want to calculate the direction of maximal increase for the entire domain: On non-border parts of the ...
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0answers
24 views

Reference Request — Laplacian on manifold without boundary

I am looking for a discussion of Laplace's equation (or something similar) on asymptotically flat or asymptotically hyperbolic manifolds without boundary. In particular, I would like to see how ...
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1answer
42 views

The second Betti number of a group

First I can't find the definition of second Betti number of a group. (Can you tell me any reference about this definition?) Also I don't know why $b_2(M)\ge b_2(G)$, where $M$ is a manifold with ...
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1answer
79 views

Is there an example of a manifold with fundamental group $\mathbb Z/3 \mathbb Z$?

I feel a little confused because I was told that there exist some manifolds with fundamental group $\mathbb Z/3 \mathbb Z$, but I can’t find an example, On the other hand, since any manifold $M$ has ...
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1answer
26 views

Finite intersection property for a locally finite open cover $\{U_j\}$ seems to contradict itself and imply that $\{U_j\}$ isn't actually a cover?

On page 84 of Adam's book on Sobolev spaces he states the 'uniform $C^m$-regularity condition' for a domain $\Omega$. This condition states that when we have a locally finite open cover $\{U_j\}$ of ...
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1answer
39 views

A manifold with boundary is locally (path) connected

I'm trying to prove that a topological manifold with or without boundaries is locally (path) connected. I think I've done the manifold without boundary part: a manifold without boundary is locally ...
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1answer
36 views

How can you determine if a point is inside a parametric 2D manifold?

Asume I have an arbitrary, parametric, closed, orientable, surface; a sphere, ellipsoid, closed cylinder, weird general cone.... If you only have access to the parametrization, how can you determine ...
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0answers
18 views

Does a topological submanifold necessarily carry the subspace topology? Is it locally closed?

This answer suggests the following definition of a topological submanifold. Definition. Let $N$ be an $n$-dimensional topological manifold. A subset $M\subset N$ is called an $m$-dimensional ...
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0answers
46 views

Show the set of mixed strategy - utility profiles such that each player is indifferent between all of her strategies is a manifold.

I am looking for help on how to solve this game theory/manifolds question. My thoughts: Informally, E is the set of mixed strategy - utility profiles such that each player is indifferent between ...
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1answer
47 views

Proving that the torus as a CW-complex is a 2-manifold

Suppose we construct the torus, $T$, as a CW-complex in the following way: Given a wedge sum of two circles that is generated by the letters $a$ and $b$, we attach a $2$-cell, $e_1^2$, on the wedge ...
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1answer
48 views

Are these definitions of a differential form equivalent?

The definition from my notes says that a differential $k$-form is a section of $\bigwedge^k T^*X \rightarrow X$, so $\omega \in \Omega^k(X)$ would be a map $\omega : X \rightarrow \bigwedge^k T^*X$ ...
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1answer
78 views

Tangent space $T_q(df(M))$ as a subspace of $T_q(T^*M)$

I have been asked to describe the tangents space $T_q(df(M))$ as a subspace of $T_q(T^*M)$ where $f\in C^\infty(M)$ and $df$ is a 1-form (or smooth section of $T^*M$). Here, $df:M\rightarrow T^*M$ ...
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0answers
38 views

Why is the Jacobi Matrix not a matrix?

In a textbook im reading about manifolds they write: Let $W \subset \mathbb{ R } ^n $ be an open set and $ F : W \rightarrow \mathbb{ R } ^ m $ a $C ^ k $ function such that for every point $ p \in ...
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10 views

Boundary of polygonal presentation homeomorphic to bouqet of circles

Suppose $P$ is a regular 2n-gon, with sides in pairs to give a surface. I want to show that the image under quotient topology of boundary of this polygon is homeomorphic to wedge of n circles. I want ...