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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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2answers
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Product of vector fields is not a vector field

Let $M$ be a manifold and $X,Y$ be vector fields on $M$. The bracket $[X,Y]:=XY-YX$ is a vector field when $X,Y$ are smooth, but why is $XY$ not a vector field when $X,Y$ are smooth? By definition, a ...
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1answer
14 views

Dimension of Manifold in Dynamical System On a Plane

I've been reading about dynamical systems on a plane and the stable and unstable manifolds that can exist there. As part of this I was reading the definition of a manifold (that a manifold is a ...
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0answers
24 views

What is really being asked by “Prove that $S^1 ⊂ R^2$ is a sub manifold”?

I'm self-studying smooth manifolds, and there is some terminology that bothers me a lot. In a lot of books, or homework questions that I looked, there are statements such as Prove that $S^1 ⊂ R^2$...
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1answer
30 views

A compact $n-1$ manifold can be embedded in $\mathbb{R}^n$ iff it can be embedded in $S^n$

I would like to ask how to show that a compact $n-1$ manifold embedded in $S^n$ can be embedded in $\mathbb{R}^n$? By Alexander duality one can show that some space can not be embedded in $\mathbb{R}^...
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14 views

Difficulty understanding how to construct different smooth structures on a manifold using the identity map

I am reading a text stating: “It is easy to give examples to show that a $C^{\infty}$ homeomorphism need not be a diffeomorphism. For any integer $n > 1$ the map $x^{2n-1}: \mathbb R \to \mathbb R$...
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1answer
17 views

Clarification regarding the definition of a Tangent Space

I am reading the introductory manifolds text by Tu. The tangent space of $\mathbb{R}^n$ at $p$, represented by $T_p(\mathbb{R}^n)$ is isomorphic to $\mathcal{D}_p(\mathbb{R}^n)$, the set of all point ...
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0answers
29 views

Computation of the push forward of vectors

I am trying to understand the push forward of a vector field by going through a specific calculation. Consider $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ given by $f(x,y,z) = (x+y+7, z-x-5)$ and ...
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0answers
20 views

Boundary of a given chain

1) To prepare learning about integration on manifolds, I am trying to find the boundary of a chain given as follows: $\sigma(t_1, t_2) = (1, t_1, t_2) - (0, t_1, t_2) + (t_1, 0, t_2) + (t_1, t_2, 1) -...
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0answers
20 views

Uniqueness of integral curves starting on boundary?

I'm only aware of ODE uniqueness on open subsets of $R^n$, which is useful when we want to prove uniqueness of integral curves on a smooth manifold. Suppose we have an interal curve $f:[0,\delta]\to ...
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1answer
39 views

Complete regularity of infinite dimensional manifolds

A Hausdorff topological space $X$ is called a manifold if $X$ is locally homeomorphic to a locally convex topological vector space. J. Eells, Jr. asserts that every manifold is completely regular (p. ...
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14 views

For $\Phi(x, y)=x^{2}-y^{2}$ can $\Phi^{-1}(0)$ be given a topology and a smooth str. s.t it is an immersed sub manifold?

In the book of Lee, introduction to smooth manifolds, at page 123, it is asked that \begin{array}{l}{\text { Let } \Phi : \mathbb{R}^{2} \rightarrow \mathbb{R} \text { be defined by } \Phi(x, y)=x^{...
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1answer
35 views

if a tangent at a point of a manifold taken as a function is zero at two points, it is zero at all points

Let $M$ be a manifold and $m\in M$. We call $t$ a tangent at $m$ if for every pair $(f,g)$ of smooth real functions defined in a neighborhood of $m$ and for every pair $(a,b)$ of real numbers, we have ...
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0answers
26 views

For which values of $a$, is ${M_{a}=\left\{(x, y) : y^{2}=x(x-1)(x-a)\right\}}$ a sub manifold?

In the book of Lee, introduction to smooth manifolds, in page 123, it is asked that \begin{array}{c}{\text { For each } a \in \mathbb{R}, \text { let } M_{a} \text { be the subset of } \mathbb{R}^{2} ...
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0answers
10 views

Is my understanding about the smooth structure on tangent bundle accurate?

I'm not sure if I understood the smooth structure on tangent bundle of a smooth manifold, so my question is whether my understanding is correct, or not, so let me explain what I've understood so far. ...
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1answer
29 views

Covering dimension of boundary of compact subset of $\mathbb{R}^n$

Let $X$ be a compact subset of $\mathbb{R}^n$, with the inherited Euclidean topology. Does it follow that $\dim_{cov}(\partial X)\leq n-1$? I would be happy to have a reference for that, if it is ...
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19 views

Please give a counterexample to show the composition of maps of constant rank need not have constant rank [on hold]

Please give a counterexample to show the composition of maps of constant rank need not have constant rank.
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1answer
20 views

Lie algebras of infinite dimensional Lie groups

I have to work with Lie algebras of some infinite dimensional 'Lie groups' (e.g. $\Omega SL_2(\mathbb{C})$) but i'm not sure on how to approach infinite dimensional groups, for loop group it is not so ...
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1answer
34 views

Given a smooth map $g: M\to R^+$, show that there is a smooth function $f:M → R^+$ such that $f(x) < g(x)$ for all $x ∈ M$.

I'm trying to prove the following proposition. Let M be a smooth manifold and let $g:M→R^+$ be a continuous function. Show that there is a smooth function $f:M→ R^+$ such that $f(x) < g(x)$ ...
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1answer
20 views

For any k-dim. subspace $ L$ of $T_p M$, can we find a sub manifold, say $R$, of $M$ containing $p$ s.t $T_p R = L$

Let $M$ be an $n$ dimensional manifold, and $S\subseteq M$ be a k-dim. sub manifold of $M$, where each is in fact a smooth manifold to be precise. We know that $T_p S$ is a k-dim. subspace of $T_p M$....
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1answer
34 views

$(0,3)$ as a non-Hausdorff smooth manifold

Consider the interval $(0,3)$ and let the neighborhoods of the points in $(0,3)-\{1,2\}$ be as those in the real line, while the neighborhoods of $i=1$ or $2$ are defined to be $(i-\epsilon,i] \cup (2,...
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0answers
11 views

Linear approximation non-zero implies surjectivity?

Consider $M$ a smooth manifold, and $g:M\to\mathbb{R}$. I want to prove the following equivalence: $dg_x:T_xM\to\mathbb{R}$ is surjective, if and only if $dg_x\neq 0$. That surjective implies $...
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1answer
20 views

Are real linear maps of smooth sections locally determined?

Let $ \pi_{ 1 } \colon E_{ 1 } \to M $ and $ \pi_{ 2 } \colon E_{ 2 } \to M $ be smooth vector bundles (of finite rank) over a smooth manifold $ M $, and consider a map $ T \colon \Gamma ( E_{ 1 } ) \...
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0answers
21 views

Question related to orientation on a arbitrary oriented manifold

This is a section from Loring Tu's book Introduction to Manifolds page 244 Second Edition. My question is as follows: Towards the end of the text in the image he says that an oriented manifold can be ...
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0answers
19 views

Condition for Riemannian distance to be equal to metric distance

If $M$ is a metric space than it is a topological space and if it is locally homeomorphic to to $R^n$ we say that it is a manifold and if we equipped this manifold with a inner product $g_p$ on ...
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1answer
38 views

How can we apply generalized Stokes' theorem to a non-oriented manifold with boundary?

I do not really know much about the boundary of non-oriented manifold. A boundary of oriented manifold, if it exists, has a sign. If you reverse the orientation, the boundary picks up an extra ...
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1answer
50 views

Is the differential at a regular point, a vector space isomorphism of tangent spaces, also a diffeomorphism of tangent spaces as manifolds?

Note: My question is not "If $f$ is a diffeomorphism, then is the differential $D_qf$ an isomorphism?" My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. I didn't study much of ...
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0answers
39 views

If G is a group homeomorphism, M manifold then M/G is a manifold.

I have this question, Let $M$ be a manifold and let $G \subset Homeo(M)$ be a group acting on $M$. Suppose that this group action is properly discontinuous and free prove that the quotient space $M/G$ ...
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0answers
9 views

how to find form of $h(x)$ in reduction to centre manifold

given the system $\dot{x} = y - x - x^2$ and $\dot{y} = x - y - y^2$ I can find that the centre subspace is spanned by $E^c = [1,1]^T$ It says the centre manifold can be expressed as $y=h(x) = x + ...
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0answers
21 views

Use clutching to show $TS^2=\mathbb{L}_{+1}^{\otimes 2}$

I'm looking for a way to show $TS^2=\mathbb{L}_{+1}^{\otimes 2}$. I'm told I should use a clutching construction, which I have very little understanding of (I've just looked at the part Karoubi's book ...
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0answers
12 views

Tangent Vectors : $\gamma\sim \gamma|_{[0,\epsilon)}$?

If $p\in \operatorname{Int} M$, given a smooth curve $\gamma: (-\epsilon,\epsilon)\to M$ such that $\gamma(0)=p$, then $\gamma|_{[0,\epsilon)}:[0,\epsilon)\to M$ is also a smooth curve such that $\...
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2answers
54 views

Why is the image of the map $f(x) =(sin(x),1- cos(x))$ for $x\in [0, 2\pi)$ and $f =\operatorname{id}$ otherwise not a submanifold of $\mathbb{R}^2$?

Consider the map $f: (-\infty, 2\pi) \to \mathbb{R}^2$ s.t $f(x) = (sin(x), 1- cos(x))$ for $x\in [0, 2\pi)$ and $f(x) = (\operatorname{id}(x), 0)$ for the rest of its domain. In the book of ...
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1answer
49 views

Is $L^{n}$ normal, where $L$ denotes the closed long ray?

1.I am trying to prove that $L^{n}$, the $n$-$th$ product of closed long ray is normal, so that I can apply Tietze extension theorem to its closed subset and prove something else. I think I am able to ...
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1answer
30 views

Differential and manifold, concrete example of a calculation

I having some problem for computing the differential of the function $f : \mathbb{T}^2 \rightarrow \mathbb{S}^2$ defined as the quotient of the function $F$ from $\mathbb{R}^2$ to $\mathbb{S}^2$ by : ...
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1answer
39 views

Cover of Shifrin Multivariable Mathematics with Manifolds?

Just curious, anyone knows what this image on the cover of Shifrin's textbook is? It doesn't seem to be a manifold.
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33 views

differential 'df' as a tensor.

In my text book there is question which says: If f is smooth real valued function on a manifold M then df is a) linear map on b)a co-vector on M c) (0,2) tensor d)(1,0) tensor. Since df at a point is ...
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0answers
48 views

Can the following lemma about HEP be generalized?

If $(X,A)$ has homotopy extension property (HEP), $A$ closed in $X$, then so does $(X\times I,X\times\partial I\cup A\times I)$ where $I$ is the unit interval. Is this still true if we replace HEP by ...
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1answer
34 views

A lemma used to prove the classification of closed 1-manifolds

In our lecture notes, we have the following lemma which is afterwards used to prove the classification of closed 1-manifolds (Compact Hausdorff spaces every point of which has a neighbourhood ...
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0answers
13 views

Prove that tangent space to a submanifold is the kernel of derivative

The definition of sub-manifold in my course is: A set $M\subset\mathbb{R}^n$ is a $k$-dimensional submanifold if for each $x_0\in M$, there exists an open neighbourhood $\Omega$ of $x_0$ in $\mathbb{R}...
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0answers
34 views

Smooth proper map $f:M\to [0,\infty)$ where $M$ is a connected manifold

I am attempting the following exercise: Let $M$ be a connected manifold. Show that there is a smooth proper map $f:M\to[0,\infty)\subset\mathbb{R}$. Surely I can just take $f$ to be the zero map; ...
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0answers
8 views

Approximating a discrete set of points with non-uniform spacing

I am trying to model a stochastic process which has variable mass dependent on its position. One way I have found to effectively achieve this is by deforming the manifold of the space that it lies in. ...
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1answer
19 views

Homology condition - bounding a disk in a handlebody?

Suppose that $\gamma_1,...,\gamma_n$ are a set of disjoint simple closed curves on a closed orientable surface $\Sigma$ that all bound disks in some handlebody $H$ with $\partial H = \Sigma$. Let $\...
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0answers
22 views

Embedding dependent vector space?

Consider $i:[a,b]\to \mathbb{R}$, defined in the simplest manner possible: $x\mapsto i(x)=x\in\mathbb{R}$. My first question would be: is this continuous? Studying topology, I realized that as it ...
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1answer
30 views

Show that there are at least two points on a manifold to which a vector is normal

Let $M \subset \mathbb R^3$ be a $2$-dimensional manifold which is also a compact set. And let $v \in \mathbb R^3$ be a vector which satisfies $ ||v|| = 1$. The task is to prove that there are at ...
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0answers
54 views

Prove existence of a Heteroclinic Orbit

How would you go around proving that there exists a heteroclinic orbit between two equilibria ( in the problem I'm trying to solve, one is a stable node(say n) and the other is a saddle (say s))? I ...
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1answer
15 views

Homotopy and connection with tangent bundle of a manifold?

Let $x$ be a point of topological space $X$, and let $\omega$, $\tilde{\omega}$ be two loops based at $x$; $\omega, \tilde{\omega}: I \to X$, with $I$ being the closed interval $[0,1]$. We say that $\...
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1answer
21 views

A manifold is covered by a union of a countably many charts

Let $M \subset R^n$ be a $k$-dimensional manifold. I want to prove that there exist a countable union of maps $r_i: V_i \to M$ such that $M = \bigcup_{i=0}^{\infty} r(V_i)$. We say that $r_i: V_i \to ...
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1answer
29 views

How to show $x-y^2+z^2=0$ and $y^2+z^2<4$ is a smooth manifold?

If $L$ is the set in $\mathbb{R}^3$ defined by $x-y^2+z^2=0$ and $y^2+z^2<4$ how would I show that $L$ is a smooth manifold?
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0answers
18 views

How to use the parameterization to find volume of ellipsoid?

If I have the ellipse in $\mathbb{R}^2$ to be defined as $x^2+4y^2=1$ with the parameterization $\gamma: \mathbb{R}^1 \rightarrow \mathbb{R}^2$ defined as $\gamma(t)=cos(t),4sin(t)$ how would I set up ...
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0answers
10 views

Prove that a union of countably many null sets in a manifold is a null set in it

Let $A \subset R^n$ be a $k$-dimensional manifold. We say that $E \subset A$ is a null set in $A$, if for every map $r:V \to A$ the set $r^{-1}(E)$ is a null set. Now let $E_1, E_2, ...$ be a ...