Questions tagged [manifolds]
For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
1
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0answers
11 views
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 ...
0
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0answers
18 views
$k$ 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^{...
0
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1answer
31 views
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 ...
2
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2answers
21 views
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 \...
0
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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\...
0
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0answers
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 ...
1
vote
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-...
1
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2answers
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 ...
0
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0answers
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 =...
1
vote
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 $\...
0
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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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0answers
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 ...
0
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0answers
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:
$$SU(5) \supset SU(3) \...
0
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0answers
26 views
Proving that $x^2+2y^2+3z^3 = 1$ is an embedded manifold
I am working on the following exercise:
Consider $S = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + 2y^2 + 3z^3 = 1 \text{ and } z>0\}$
Show that $S$ can be parametrised as a graph of a function from an ...
0
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1answer
39 views
Show that $g^{-1}\frac{dg}{dt} \in \mathfrak{g}$
Let's $G$ be a Lie Group with its respective Lie Algebra $\mathfrak{g}$. I would like to show the next statement:
If $g(t)$ is a smooth curve in $G$, then $g^{-1} \frac{dg(t)}{dt} \in \mathfrak{g}$
...
2
votes
1answer
49 views
How to show that this space is not a manifold?
Define $X = (\mathbb{R}^2 \backslash \{0\})/\sim$ with the quotient topology induced by the projection $\pi: \mathbb{R}^2\backslash \{0\} \rightarrow X$ where the equivalence relation $\sim$ on $\...
1
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2answers
29 views
Is a manifold $N$ smoothly embedded in a manifold $M$ of the same dimension open in $M$?
Consider a manifold smooth manifold $N$ smoothly embedded in another manifold $M$ of the same dimension. Is it true that $N$ is open in $M$? I think this is true, due to the open mapping theorem.
If ...
0
votes
0answers
19 views
The essential spectram of the compact manifold
I heard that the essential spectrum of the compact Riemannian manifold is empty.
Why does this hold?
I would appreciate if you could teach me about this.
1
vote
0answers
23 views
What is smooth structure on the exterior algebra of cotangent bundle?
Let $M$ be a smooth manifold. Let $(T^*M, \pi, M)$ be cotangent bundle, where
$$ T^*M := \bigsqcup_{m\in M} T^*_m M$$
is a set equipped with corresponding topology $O_{T^*M}$ and smooth structure $...
2
votes
1answer
50 views
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 ...
4
votes
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 ...
0
votes
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, \...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
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$.
...
0
votes
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$ ...
0
votes
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 ...
0
votes
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 ...
1
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0answers
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,...
1
vote
2answers
49 views
$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 ...
0
votes
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-...
1
vote
2answers
52 views
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}...
0
votes
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 $\...
4
votes
2answers
85 views
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 ...
0
votes
0answers
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, ...
1
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0answers
35 views
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\}$....
1
vote
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 \...
1
vote
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 $...
0
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0answers
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) \...
1
vote
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 = \...
0
votes
0answers
14 views
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
$$
T_{r , s}(...
2
votes
1answer
41 views
A topological problem in defining the tangent bundle
In one of the questions of my homework, tangent bundles are defined as in the picture. So I was wondering how to prove the openness of $\pi^{-1}(U)$, since the question only states that $\pi^{-1}(U)$ ...
0
votes
2answers
36 views
smooth curve on a manifold
Say i have the smooth manifold $ M=\Re^2 $, and a smooth curve $\gamma:\Re \to M$ with $\gamma(t)=(t,t)$.
Can i draw this curve to the manifold without the use of any chart?
Maybe it sounds a silly ...
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0answers
61 views
Is the special linear group a submanifold?
The book I'm reading defines the special linear group as
$$\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
$...
4
votes
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
0answers
20 views
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 ...
