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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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Bounded 4-manifold signature

My understanding is that the same 3-manifold may have more than one bounded 4-manifold depending on what particular Heegaard decomposition you use to construct your 3-manifold. I would like to be ...
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$k$ times differentiable but not $C^k$ manifold

I cannot find the notion of $k$ times differentiable manifold with non-continuous $k^{th}$ differential, i.e. a manifold with charts having $k$ times differentiable transition maps, but where the $k^{...
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1answer
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Does a manifold of dimension one has curvature?

Recently I have seen an interesting answer to an "obvious" question. That is "why can we pull a curve back into a line"? And the answer is "because a manifold of dimension one has no curvature". So I ...
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Are these conditions equivalent to the definition of regular coordinate ball?

In page 15 of Lee's book "Introduction to Smooth Manifolds", there's a paragraph as follows: We say a set $B\subset M$ is a regular coordinate ball if there is a smooth coordinate ball $B'\supset \...
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1answer
19 views

Smooth Urysohn function having $0$ as regular value

Let $M$ be a manifold and $C_0,C_1$ be two disjoint closed subsets of $M$ , then smooth Urysohn Lemma says that there exists a smooth function $f:M\rightarrow [0,1]$ such that $f(C_0)=\{0\},f(C_1)=\{1\...
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50 views

Characteristic classes of unit sphere bundle

Let $M$ be a smooth manifold and $\xi:E\to M$ a real vector bundle over $M$. Suppose we fix a metric $g$ on $E$ so that we can define the unit sphere bundle $S(E)\to M$. How are the characteristic ...
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1answer
32 views

Time-dependent manifolds

I'm studying Classical Mechanics, and in some cases the particles in the study are constrained to move in a certain manifold, which changes with time. I've looked for bibliography about time-...
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45 views

Is this surface a compact manifold?

Let $\Omega \subset \mathbb R^3$ denote an open, bounded and connected set with $C^2-$regular boundary $\Gamma=\partial \Omega$. Is it true that $\Gamma$ is a compact manifold? Disclaimer: It 's ...
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16 views

Projection On Standard Coordinates

I was Thinking on problem 1.3.9 from Guillemin and Pollack Differential Topology. There's a Hint Which I had difficulty to prove for some reason. Suppose that you have a subspace $ V $ with $ dim V =...
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1answer
56 views

Let $\mathbb H$ be algebra of quaternions and let $S$ be the group of unit quaternions. What does it mean for a point to be tangent to $\mathcal S$?

Let $\mathbb H$ be the algebra of quaternions and let $\mathcal{S} \subset \mathbb H$ be the group of unit quaternions. Show that if $p \in \mathbb H$ is imaginary, then $qp$ is tangent to $\...
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1answer
44 views

$\bigcup_{i \in I} M_i$ is a manifold when $\forall j \in I, M_j \cap \overline{\bigcup_{i \neq j} M_i}= \emptyset$

Let $M_i, i \in I$ be a family of $d$-dimensional manifolds in $\mathbb{R}^p$ so that $$\forall j \in I, M_j \cap \overline{\bigcup_{i \neq j} M_i}= \emptyset$$ How can I show that $\bigcup_{i \in I} ...
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31 views

Tangent vector to a curve as a function

Carmo's book "Riemannian Geometry" defines a differentiable manifold and tangent vector as follows respectively. Why is the tangent vector defined as a function? Is this because in going from the ...
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19 views

Lie group embeddings $SU(5) \supset SU(3) \times SU(2) \times U(1)?$

Does the special unitary Lie group $SU(5)$ contains $SU(3) \times SU(2) \times U(1)$ as a subgroup? Can one show which of the following embeddings are possible rigorously: $$SU(5) \supset SU(3) \...
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Proving that $x^2+2y^2+3z^3 = 1$ is an embedded manifold

I am working on the following exercise: Consider $S = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + 2y^2 + 3z^3 = 1 \text{ and } z>0\}$ Show that $S$ can be parametrised as a graph of a function from an ...
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1answer
39 views

Show that $g^{-1}\frac{dg}{dt} \in \mathfrak{g}$

Let's $G$ be a Lie Group with its respective Lie Algebra $\mathfrak{g}$. I would like to show the next statement: If $g(t)$ is a smooth curve in $G$, then $g^{-1} \frac{dg(t)}{dt} \in \mathfrak{g}$ ...
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1answer
49 views

How to show that this space is not a manifold?

Define $X = (\mathbb{R}^2 \backslash \{0\})/\sim$ with the quotient topology induced by the projection $\pi: \mathbb{R}^2\backslash \{0\} \rightarrow X$ where the equivalence relation $\sim$ on $\...
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2answers
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Is a manifold $N$ smoothly embedded in a manifold $M$ of the same dimension open in $M$?

Consider a manifold smooth manifold $N$ smoothly embedded in another manifold $M$ of the same dimension. Is it true that $N$ is open in $M$? I think this is true, due to the open mapping theorem. If ...
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19 views

The essential spectram of the compact manifold

I heard that the essential spectrum of the compact Riemannian manifold is empty. Why does this hold? I would appreciate if you could teach me about this.
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What is smooth structure on the exterior algebra of cotangent bundle?

Let $M$ be a smooth manifold. Let $(T^*M, \pi, M)$ be cotangent bundle, where $$ T^*M := \bigsqcup_{m\in M} T^*_m M$$ is a set equipped with corresponding topology $O_{T^*M}$ and smooth structure $...
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Fundamental Domain and Transversality of Vector Field in Ordinary Differential Equations

Consider the ODE \begin{align} \dot x = f(x) \tag{1} \end{align} Let $x_0$ be a hyperbolic fixed point of $(1)$. Let $V$ be a neighborhood of $x_0$ in $W^s(x_0)$, where $W^s(x_0)$ is the stable ...
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4answers
251 views

Is the torus with one hole homeomorphic to the torus with two holes?

I would like to understand why the torus with one hole is not homeomorphic to the torus with two holes. I have a very basic understanding of the concepts (I know what an homeomorphism is but not much ...
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1answer
24 views

Injective Map $\gamma \in C^{\infty}(V, \mathbb{R}^{n+k})$ with Surjective Differential is Homeomorphism

I'm looking for a proof of following theorem: Let $M \subset \mathbb{R}^{n+k}$ a $n$-dimensional submanifold and $V \subset_{\text{ open}} \mathbb{R}^n$ open set. Let $\gamma \in C^{\infty}(V, \...
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1answer
31 views

Why is this definition of differential for manifolds correct?

The book we are using in class is Frank Warner Foundation of Differential Manifold and Lie Group. Let $M,N$ be two smooth $d$-dim manifold, the differential of a $C^\infty$ function $\phi:M\rightarrow ...
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1answer
24 views

Computing differentials on manifolds

I have been studying the properties of the differential operator on manifolds. Given differentiable manifolds $M,~N$ and a function $f \in C^{\infty}(M,N)$, we define the differential at a point $p\in ...
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24 views

How to proof a set is not a embedded manifold

I got stuck on one of my Maths exercises and I hope you can help me out to proceed: Consider the set $S = \{(x,y)\in \mathbb R^2: x^2=y^2\}$. Show that S is not an embedded manifold in $\mathbb R^2$. ...
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1answer
23 views

$U(n)$ and $SU(n)$ are connected smooth submanifold of $M_{n,n}(\mathbb{C})$

How can I prove that $U_n(\mathbb{C})$ and $SU_n(\mathbb{C})$ are smooth submanifolds of $M_{n,n}(\mathbb{C})$ ? I know that given the manifold $X$, $Y$ is a smooth submanifold of $X$ if $\forall k$ ...
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0answers
19 views

Finding or fitting a manifold to a set of points in a euclidean space

The question I'm going to ask is rather a vague one. I try my best to describe the question as best as I can. Because of the generality or maybe vagueness of my question, I'm not looking for an exact ...
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1answer
21 views

Every banach manifold is locally arcwise conneted [closed]

I found this inside a proof of another fact. I've been trying to find any result to prove it, but I couldn't find any. Does someone know this result or how to prove it? I'm working with a manifold ...
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42 views

Brieskorn 4-manifold

As continuation of this question consider subset $N^4=\{Re(x^p+y^q+z^r)=0:x,y,z\in\mathbb C, |x|^2+|y|^2+|z|^2=1\}$ in sphere $S^5$. It is 4-manifold in sphere which contains Brieskorn $\Sigma(p,q,...
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2answers
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$M =\{(t, \vert t \vert) \text{ }\vert t \in \mathbb{R} \} $ isn't a Smooth Submanifold

I want to show formally that $$M =\{(t, \vert t \vert) \text{ }\vert t \in \mathbb{R} \} $$ is not a smooth $C^{\infty}$-submanifold of $\mathbb{R}^2$. My attempts: Intuitively it's clear that ...
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1answer
35 views

SO(n) as a manifold

I cannot find some basic information on $SO(n)$ ($n$ general, not just 3) as a manifold: what is the geodesic distance between two matrices, what are the eigenfunctions and eigenvalues of the Laplace-...
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Unit Spheres, Isotopies, and Homotopies

I'm really struggling with the following problem. Let $f:S^{2}\to\mathbf{R}^{3}$ be the embedding of the unit sphere, and let $E$ be the ellipsoid $E=\{(x,y,z)\in\mathbf{R}^{3}~:~4x^{2}+9y^{2}+z^{2}...
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1answer
16 views

For a locally euclidean space X, what is the domain of the coordinate map for a chart?

Given a locally Euclidean space X of dimension n and a point $p \in X$, by definition there exists a neighborhood $U \subset X$ and a homeomorphism $\phi$ such that $\phi (U)$ is an open subset in $\...
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Can the torus and the Klein bottle be thought as $\mathbb{R}^2/G$ with $G$ a finite group acting freely

I have recently proved the following exercise, Let $G$ be a finite group acting freely on a (compact) topological manifold of dimension $n$. Then, $X/G$ is a (compact) manifold of dimension $n$. I ...
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19 views

Source For Partitions of Unity Problems

I’ve realized that I need to work more with partitions of unity, but unfortunately there are not a lot of problems in the partitions of unity section in Lee which enable you to practice using them, ...
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A proof of showing some flow is global

Given a flow $\Phi_i(t,p)$ defined on $A_i=\mathbb{R}\times E$, and $f$ is a proper map from $E$ to $\mathbb{R}^n$, s.t. $f\Phi_i(t,p)=f(q)+e_it$. And define $J_q$ by $A_i=\cup_{q\in E}J_q\times\{q\}$....
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1answer
76 views

A special change of coordiantes of a Vector Field

Let $X:\mathbb{R}^3 \to \mathbb{R}^3$ be a smooth vector field and $0 \in \mathbb{R}^3$ a regular point (i.e. $X(0)\neq 0$), it is well known that there exists a diffeomorfism $\varphi:V_0 \subset \...
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1answer
32 views

Another representation of $S^n$ as a quotient of disk

Let $D^n\subset \mathbb{R}^n$ be the subset consisting of those points $(x_1,\dots,x_n)\in \mathbb{R}^n$ such that $x_1^2+\dots+x_n^2\leq 1$ and let $S^{n-1}\subset D^n$ be the subset of those points $...
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34 views

Stiefel Whitney classes on the simplex or the simplicial complex

The Stiefel Whitney classes of the base manifold $M$ are characteristic class as $$ w_j(M) \in H^j(M,\mathbb{Z}_2), $$ Puzzle: How do we write $$ w_1(M) \in H^1(M,\mathbb{Z}_2) $$ $$ w_2(M) \...
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1answer
36 views

Lie bracket of vector fields is a vector field

Let $X$ and $Y$ two vector field on the manifold $M$ (dim($M$)= $m$). Show that the Lie bracket $[X, Y] := XY - YX $ is a vector field on $M$. I tried to compute using local coordinates, so for $X = \...
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A topology in a (concrete) vector space for tensors and manifolds.

Let $M$ be a (real) $n$-dimensional differentiable manifold. A $(r , s)$ type tensor on $M$ is defined by a differentiable map $t : M \to T_{r , s}(M)$ such that $\pi \circ t = Id$, where $$ T_{r , s}(...
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1answer
41 views

A topological problem in defining the tangent bundle

In one of the questions of my homework, tangent bundles are defined as in the picture. So I was wondering how to prove the openness of $\pi^{-1}(U)$, since the question only states that $\pi^{-1}(U)$ ...
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2answers
36 views

smooth curve on a manifold

Say i have the smooth manifold $ M=\Re^2 $, and a smooth curve $\gamma:\Re \to M$ with $\gamma(t)=(t,t)$. Can i draw this curve to the manifold without the use of any chart? Maybe it sounds a silly ...
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61 views

Is the special linear group a submanifold?

The book I'm reading defines the special linear group as $$\mathrm{SI}(n) = \{X\in \mathbb{R}^{n\times n} \mid \det X = 1\}$$ I expanded the determinant of X as suggested by the book and I found that $...
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1answer
42 views

Closed oriented manifold with middle Betti is one with odd degree.

The rational cohomology ring of complex projective plane $\mathbb{CP}^{2}$ is truncated polynomial ring $\frac{\mathbb{Q}[X]}{(X)^{3}},\,\,deg(X)=2$. In this case, the degree of a generator is 2. Is ...
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1answer
37 views

(pseudo-)Riemannian manifolds and global coordinates

I have a question about (pseudo-)Riemannian manifolds. In general relativity one often assumes that there exists a global set of coordinates describing the whole manifold. My question is: Does there ...
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1answer
19 views

Closed oriented even dimensional manifold with only three non-zero Betti numbers.

The complex and quaternionic projective planes are the examples of a closed oriented even dimensional manifold with exactly three non-zero Betti numbers. For more example see the paper ''Rational ...
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65 views

Where did the term manifold come from?

I first heard the word manifold in regards to an exhaust manifold of a car. Wikipedia says the term was introduced by Riemann. Riemann lived 1826-1866 and the combustion engine was also developed ...
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27 views

Why is the submanifold a Nullset?

I tried to explain myself why a sumbanifold of the dimension $k<n$ is a Nullset in $\mathbb{R}^n$. May this be explained by the fact, that the submanifold is locally coverable by the pictures of ...
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Open after being projected

Can someone explain why a open set lets say of $\mathbb{R}^{n+l}$ is still open after being with its first n coordinates projected on the $\mathbb{R}^n$? Thank you for your help. My informal argument ...