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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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Manifolds and covering maps [duplicate]

I am studying topological manifolds; I know that if $M$ is an $m$-topological manifold and $p:M\rightarrow N$ is a covering map, then $N$ is an $m$-manifold. I know how to prove the existence of local ...
user1255055's user avatar
1 vote
1 answer
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M is composed of line segments connecting ellipse to $(0,0,0)$ Calculate integral $\int_M \sqrt{x + 3z}\ d \lambda_2$ over those. Almost done.

I found such an exercise among my set of exercises preparing for exams and I have no idea how to solve that. Every point of ellipse $\{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1, x + z = 1 \}$ is ...
thefool's user avatar
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0 answers
35 views

Immersion and Embedding [duplicate]

I am reading "Manifolds, Vector Fields,and Differential Forms" by Gross. There we defined the notion of immersion and embedding as following: A smooth map $F:M\to N$ is an immersion if $\...
Schrödinger's cat's user avatar
1 vote
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29 views

Show that tangent spaces of surface are parallel to the z-axis

I've been reviewing some past exam questions for my intro to differential geometry course and I'd like your opinions on this one: Show that $x^2+y^2-z^2=1$ defines a regular surface S. After that, ...
MathyUni's user avatar
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2 answers
90 views

Tangentbundle of submanifold of $\mathbb{R}^n$

Let $M\subseteq \mathbb{R}^n$ be a $k$-dim submanifold of $\mathbb{R}^n$, and I want to prove that the tangent bundle of $M$ is a submanifold of $\mathbb{R}^n$. The idea is: Since $M$ is a submanifold ...
Gao Minghao's user avatar
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0 answers
24 views

Triangulations of manifolds are non-branching

Let $X$ be an $n$-manifold and let $K$ be a triangulation of that manifold. I am looking for a proof of the fact that $K$ is non-branching, which means: There is no simplex $S \in K$ of dimension $n-1$...
shuhalo's user avatar
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3 votes
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Short exact sequences of orthonormal frame bundles

Given an orthonormal $l$-frame bundle $V_l(TM)\rightarrow M$ for a smooth (oriented) $d$-manifold $M$, there are short exact sequences on homotopy groups $\mathbb{Z}\rightarrow \pi_{d-l}(V_l(TM))\...
Tiana's user avatar
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2 votes
1 answer
67 views

Confusion on Notations of Partial Derivatives on Manifolds

I'm confused with the notations of partial derivative on manifolds in Tu's An Introduction to Manifolds.. Just to make clear the notations I'm using, what I've known and which part I'm confusing, ...
TheHan6edMan's user avatar
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47 views

Find a topological manifold that is not Hausdorff and not locally compact

Define a topological manifold as a space locally homeomorphic to $\mathbb{R}^n$. Find a topological manifold that is not Hausdorff and not locally compact. (Hint:Consider $\mathbb{R}$ with "extra ...
Ali's user avatar
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2 votes
1 answer
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If $M$ is a connected topological manifold and $p,q \in M^{o}$, then there exists a homeomorphism which maps $p$ to $q$.

I am stuck in constructing a explicit homeomorphism $f$(say) from a connected manifold $M^{o} = M \backslash \delta M$ to itself such that for $p,q \in M^{o}$ then $f(p) = q$. I know that that since $...
Dwaipayan Sharma's user avatar
2 votes
0 answers
46 views

Hodge star decomposition in non-diagonal manifold product

I'm studying differential forms and I came across the following problem. From what I learnt in another question, when a manifold can be decomposed as $X \times Y$, then the formula found there works ...
Fredrigo6's user avatar
3 votes
1 answer
93 views

What is the fundamental group of the space $X_n=SL(n, \mathbb{R})/SL(n, \mathbb{Z})$

The space of n dimensional unimodular lattices is often parametrised by the homogeneous space $$X_n=SL(n,\mathbb{R})/SL(n,\mathbb{Z}).$$ It is a manifold, as it is a quotient of a Lie group by a ...
Sundara Narasimhan's user avatar
2 votes
1 answer
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$\Lambda^n(M)$ is not isomorphic to $C^{\infty}(M)$ if M is not orientable

Let $M$ be a differentiable Manifold of dimension n. If $M$ is orientable, then there exists an $\omega \in \Omega^n(M)$ (top-degree differential form) such that $\omega(p) \neq 0$ $ \forall p \in M$. ...
Jahi02's user avatar
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Definition of a (holomorphic) differential form in the context of translation surfaces

I am a beginning graduate student with an interest in geometry, in particular, in translation surfaces. I am trying to learn from the recent text by Athreya & Masur. My biggest point of confusion ...
Steven Cripe's user avatar
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1 answer
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Question about the coordinate system on the manifold

Given an $n$-dimensional manifold $M$ and a homeomorphism $\phi:U\subset M\rightarrow V\subset\mathbb{R}^{n}$ from a patch on the manifold $U\subset M$, then we can parametrize a point $p\in U$ via ...
Andyale's user avatar
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1 vote
2 answers
133 views

Why is the dimension of this square $n \times n$ matrix given as $n^2 - 1$?

There's a hole in my knowledge. I tried plugging it by reading more of Lee's book, but I couldn't easily find why the rank of this matrix is less than the number of elements squared. I'm afraid this ...
Nate's user avatar
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2 votes
1 answer
100 views

For a compact 4-manifold, no 2-torsion in $H_1(M;\Bbb Z)$ implies no 2-torsion in $H_n(M;\Bbb Z)$ for all $n$

Let $M$ be a topological compact connected oriented 4-manifold with nonempty boundary, and suppose that each boundary component of $M$ is a rational homology 3-sphere. Is it true that if $H_1(M;\Bbb Z)...
user302934's user avatar
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2 votes
1 answer
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Identity for curvature tensor

I need to show the following idetnity, where $R$ is riemannian curvature tensor of a manifold $(M,g)$ and $x,y,z,t\in T_pM$: $$ \partial_\alpha\partial_\beta\{R(x+\alpha t,y+\beta z,y+\beta z,x+\alpha ...
Gao Minghao's user avatar
-2 votes
0 answers
56 views

Prove that a loin with it's boundaries is closed

So, I'm not very advanced in topology, but I've noticed that lines on manifolds can have interesting properties. But I don't know if and how lines are defined on general topological spaces (is extra ...
עמית חי לרמן's user avatar
1 vote
1 answer
61 views

If a Banach manifold satisfies the Heine-Borel property, then does it have finite dimension? [closed]

Suppose $C$ is a topological Banach manifold, that is also a closed convex subset of a Banach space $E$, also, $C$ satisfies the Heine-Borel property: Every closed and bounded (with respect to the ...
Raul Fernandes Horta's user avatar
2 votes
0 answers
36 views

Showing that $\theta:U \to |\mathcal{F}|, x\mapsto(f\mapsto f(x))$ is a homeomorphism

Suppose that $\mathcal{F}=C^\infty(U)$ is the $\Bbb R$-algebra of infinitely differentiable real-valued functions on an open set $U \subset \Bbb R^n$ and denote $|\mathcal{F}|$ the space of maps $C^\...
Iman's user avatar
  • 171
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0 answers
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Confusion about integration in manifolds

I'm reading through Sean M. Carroll's text on General Relativity. In section 2.10, he explains that the volume element $d^nx$ transforms as a tensor density. Consequently, to get a "tensorial ...
Aidan Beecher's user avatar
2 votes
0 answers
30 views

Surjective local diffeomorphism with finite fibers is a covering map? [duplicate]

Let $E, M$ be connected manifolds and $\pi: E \to M$ a surjective local diffeomorphism with finite fibers. Is $\pi$ necessarily a covering map? I know that this is true if $E$ is compact, or more ...
J. C.'s user avatar
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2 votes
1 answer
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Can this closed form not exist on the sphere?

This is part (b) to a problem that I solved part (a) for a while back. I finally had time to complete part (b), and I wanted to know if this is a valid answer (proved correctly): Problem Statement: ...
Nate's user avatar
  • 894
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0 answers
37 views

Proof of section is smooth iff component functions are smooth

I'm trying to understand the proof of this lemma in "An introduction to manifolds" 2ed by Loring W.Tu. (p. 138) Lemma 12.11 Let $\phi:E|_U \longrightarrow U \times \mathbb{R}^r$ be a ...
MLe's user avatar
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Lee Smooth Manifolds Theorem 6.24 proof

I'm currently reading the book Smooth Manifolds by John M.Lee. There is something unclear to me in the proof of theorem $6.24$, the Tubular Neighborhood Theorem. The proof goes as follows. First, if $...
Mark's user avatar
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Reference Request: Is Advanced Calculus enough for someone like Tu's or Lee's introductory books on (Smooth) manifolds?

Basically title. At my university there is a course (Graduate Differential Topology) that uses Lee's introduction to smooth manifolds and Tu's introduction to manifolds, and the listed pre requisites ...
baslerbuenzli's user avatar
1 vote
0 answers
55 views

Is this the correct value for integrating this differential form over a sphere?

I posted this problem but made a mistake. I redid my work and got another sensible looking answer (the volume of the unit sphere). Does this look like a correct solution? This isn't an assigned ...
Nate's user avatar
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0 answers
31 views

Equivalence between two definitions of a Seifert Fibered Homology Sphere

I am reading Savaliev's Invariants for Homology 3-spheres. Here, he defines the Seifert Fibered Homolgy Sphere as $\Sigma(a_1,...,a_n) = V_B(a_1,...,a_n) \cap S^{2n-1}$ where $V_B(a_1,...,a_n) = \{b_{...
user13121312's user avatar
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0 answers
39 views

Why a double cone is not a $3$-dimension manifold

I am trying to understand why a double cone is not a manifold. A double cone refers to 2 cones in $R^3$ joined at the vertex. Formally, it is given by $$ 𝑆={(𝑥,𝑦,𝑧)\in R^3:𝑧^2=𝑥^2+𝑦^2} $$ I ...
Momin Ahmed's user avatar
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2 answers
43 views

Each elements in Locally finite precompact open cover of a topological manifold has at most finite intersection with others

The problem is from John M Lee's Introduction to Smooth Manifold:problem 1.4.(see below image) I have checked similar post regarding this problem from elsewhere of this site. However, I still find it ...
mikeqwertyuiop's user avatar
6 votes
1 answer
89 views

Can the twice-punctured plane be given a homogeneous metric?

A homogeneous metric on a space $X$ is one for which the isometry group acts transitively on its points (for all $x,y\in X$, there is an isometry $\phi$ of $X$ such that $\phi(x)=y$). If we repeatedly ...
volcanrb's user avatar
  • 3,054
1 vote
1 answer
57 views

Open subset of paracompact manifold is paracompact?

I assume a smooth manifold is paracompact (including Hausdorff) instead of Hausdorff and second countable. We can show that a connected topological manifold is paracompact iff it's second countable. ...
wsz_fantasy's user avatar
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0 answers
33 views

What is the theory of the measure of a line in a plane

I have been taught how to measure the length of a line in a plane, in typical situations encountered in physics; for instance, by integrating $dl = \sqrt{dx^2+dy^2}$ along the line, parametrized in ...
Twizzle's user avatar
  • 89
0 votes
0 answers
38 views

What's the continuous $\mathbb C^0$ version for immersion? I'm trying to see if topological embeddings are 'topological immersions.'

A 'smooth embedding' $f: M \to N$ between smooth ($m$,$n$)-manifolds ($M$,$N$) is a smooth map that is both a smooth immersion and a topological embedding, which is simply defined as that the range-...
BCLC's user avatar
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0 votes
0 answers
61 views

Can locally Euclidean Hausdorff 2nd countable topological space always be made into topological manifolds (as in continuous or $C^0$)?

For any locally Euclidean Hausdorff 2nd countable topological space $M$, can $M$ always be made into a topological manifold (not necessarily of some uniform dimension $n$ ... but if ever I assume ...
BCLC's user avatar
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0 votes
1 answer
55 views

Do smooth manifolds care about the continuous atlas?

A smooth manifold $M$ can be defined as a pair (topological manifold $X$, smooth, as in $C^{\infty}$, atlas $\mathscr M$), where a topological manifold is defined as a locally Euclidean Hausdorff 2nd ...
BCLC's user avatar
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0 votes
1 answer
28 views

Example of Critical Path

If $F$ takes on its minimum at a path $\omega_0$, and if the derivatives $\frac{\mathrm{d}F\left(\bar{\alpha}(u)\right)}{\mathrm{d}u}$ are all defined, Milnor states in his Morse Theory that clearly $\...
一団和気's user avatar
1 vote
0 answers
31 views

Smoothness of a cylinder

I´m trying to check the smoothness property of a cylinder $D:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2\leq 1, z\in [0, H]\}$, but I´m having a problem understanding the definition of Lipschitz continuous ...
oli H.'s user avatar
  • 329
0 votes
1 answer
43 views

An analogue of the de Rham complex for vector fields

Let $M$ be a smooth manifold and let $\chi$ be its tangent bundle and $\Omega^1$ be its bundle of $1$-forms. Using the exterior algebra we can extend the space of sections of $\Omega^1$ to a ...
Zoltan Fleishman's user avatar
0 votes
1 answer
47 views

example of non-compactly supported vector field.

Definition: A vector field X on M is complete if for any p $\in$ M, there is an integral curve $ \gamma: R \to M$ such that $\gamma(0) = p$. Note: every smooth vector field over a compact manifold is ...
Hemanta Mandal's user avatar
1 vote
0 answers
23 views

Proof that the set of all doubly stochastic matrices is a manifold?

I'm trying to find the proof of a fact that the set of all doubly stochastic matrices of positive elements: \begin{equation} \mathcal{D}\mathcal{P}_{n} := \{ X \in \mathbb{R}^{n \times n}; ~ \forall i,...
mathmrk's user avatar
  • 183
1 vote
1 answer
113 views

Motivation behind using term “manifold” for manifolds? [closed]

Why Riemann picked word “manifold” for his work in that area? I read somewhere that it might have to do with geometry in more that 3 dimensions, since three numbers (x,y,z) are enough to give ...
coobit's user avatar
  • 219
0 votes
1 answer
48 views

Whitney's strong embedding theorem for Manifolds with Boundary

I have a question on a remark from wikipedia on Whitney's strong embedding theorem: The general outline of the proof is to start with an immersion $f: M\to \mathbb {R} ^{2m}$ with transverse self-...
user267839's user avatar
  • 7,499
2 votes
0 answers
57 views

Open neighborhood of a geometric knot in an orientable 3-dimensional manifold.

I was reading "Braid Groups" by Christian Kassel and Vladimir Turaev where I found the following question: Prove that an arbitrary geometric knot $L$ in an orientable 3-dimensional manifold ...
ripan sharma's user avatar
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0 answers
45 views

Manifolds and Euclidean Spaces

This might be a basic question, but it's been irking me for the past few days. The common definition of a manifold is as a second-countable, Haussdorff topoplogical space which is locally homeomorphic ...
markusas's user avatar
  • 358
1 vote
1 answer
89 views

Proof: Let $M$ be a manifold then $M\setminus p$ isn't compact.

Let $M$ be a manifold and $p \in M$. Then $M \setminus p$ isn't compact. I have seen some proofs of this with open covers and so on, but i wonder if the following proof is correct: Proof: Let $\kappa:...
Jahi02's user avatar
  • 301
0 votes
0 answers
9 views

Using Implicit Function Theorem to Find a Coordinate System near a point of a level set map

Let $F: \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function, $c \in \mathbb{R}^m$ be a regular value and $S=F^{-1}(c)$ be the embedded submanifold given by the regular level set. It is well-known that ...
TheWanderer's user avatar
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0 votes
0 answers
33 views

Show that the given stereographic projection is $\sigma(x)=u$

Let $N$ be the North pole $U_N=(0,\dots, 0, 1)$ and the South pole $U_S=(0,\dots, 0, -1)$, where both are $\in \mathbb{S}^n\subseteq \mathbb{R}^{n+1}$. Define the stereographic projection $\sigma:\...
Superunknown's user avatar
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0 votes
1 answer
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Proof of Existence of Partitions of unity in "Introduction to Topological manifolds"

I have a question reading the proof of Existence of Partitions of Unity in "Introduction to Topological manifolds" 2nd by John M.Lee. (p.114-115) The statement is: Theorem 4.85 Let X be a ...
MLe's user avatar
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