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

learn more… | top users | synonyms (1)

0
votes
1answer
261 views

Manifold learning/nonlinear dimensionality reduction for beginners

I'm a computer science graduate student. I recently discovered manifold learning. I think I understand the very basic, high-level concept of nonlinear dimensionality reduction, but I'd like a ...
4
votes
1answer
461 views

Partial Derivatives on Manifolds - Is this conclusion right?

I'm self-studying Differential Geometry and I've asked here about how to describe functions on a manifold, and now that I'm pretty sure that my conclusions about that are correct I've started to think ...
1
vote
1answer
50 views

How to build an specific atlas for the $n$-disk $D^n$

How can we build a topological atlas for the $n$-dimensional disk $D^n=\{ x\in \mathbb{R}^n \mid \lVert x \rVert \leqslant 1 \}$ as a manifold with boundary ? Specifically, how to construct the maps ...
6
votes
1answer
186 views

Describing Functions on a Manifold

Well, I have a pretty simple (probably silly and basic) doubt about how do we describe functions on a Manifold. Well, just to make clear which definitions I'm using, for now what I know is that: a ...
3
votes
1answer
220 views

Defining the coboundary map $\delta_*$ on the Sheaf Cech Cohomology groups

I'm having trouble understanding the definitions I've been reading, of what has been called an 'induced coboundary operator' or a 'connecting homomorphism' depending on what source you're reading. ...
2
votes
1answer
181 views

How to Define Product Orientations for Topological Manifolds

When working with smooth manifolds, $M^m$ and $N^n$, it is straightforward to see how orientations at points $p\in M$ and $q\in N$ (i.e. ordered bases for the tangent spaces) give rise to an ...
3
votes
2answers
74 views

$\mathbb{R}P^2$ and its lines

I have been solving some past exam questions and I came across the following question. Let $r$ and $s$ two distinct lines in $\mathbb{R}P^2$, and let $X$ the space obtained contracting $r \cup s$ to a ...
0
votes
1answer
33 views

manifold optimization

can anyone help me to solve the following optimization problem? thank you very much. min_{U} F(U) subject U'U=I_r where U is a matrix of ...
3
votes
1answer
82 views

One remark on Integration on Manifold

I was reading the book Introduction to Manifold by Loring W Tu. And I am confused with a remark Tu made in his book. I need a little bit of clarification. In Chapter 5 (differential forms), he wrote ...
4
votes
1answer
207 views

Non-vanishing vector field on $\mathbb{R}P^{2n+1}$

I'm trying to cook up a non-vanishing vector field on $\mathbb{R}P^{2n+1}$. I know that $S^{2n+1}$ admits one, namely $(x_1,\dots,x_{2n+2})\mapsto (-x_2,x_1,\dots,-x_{2n+2},x_{2n+1})$. Moreover, I ...
1
vote
3answers
848 views

Asking for a good starting tutorial on differential geometry for engineering background student.

I just jumped into a project related to an estimation algorithm. It needs to build measures between two distributions. I found a lot of papers in this field required a general idea from differential ...
1
vote
1answer
190 views

Partial Derivative on a Manifold

Compute the partial derivative of $f(x,y) = 2x + y^3$ at $a = (x_0,y_0)$. I know this looks easy but the purpose of me asking this question is to see how this question is worked as if it we were just ...
3
votes
1answer
90 views

Polar decompostion should be a diffeomorphism, right?

I seem to have gotten stuck in the mud verifying what I thought was going to be a completely straightforward fact. I would appreciate if somebody could help dig me out. Inside the $n \times n$ ...
6
votes
1answer
266 views

A problem from Spivak's Calculus on Manifolds

Notation As Spivak suggests, given $A\subset\mathbb R^n$, boundary $A$ denotes the topological boundary of $A$, i.e. $\overline A\cap\overline{A^c}$. Problem 5-3(a): Let $A\subset\mathbb R^n$ be ...
2
votes
2answers
85 views

studying compact $\partial$-$n$-manifolds via closed $n$-manifolds?

What would be counterexamples to the following statement: It is not true that any $n$-manifold with boundary is a $n$-manifold with finitely many embedded disjoint open disks removed, since that ...
4
votes
0answers
218 views

Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
6
votes
1answer
158 views

Why is $\mathbb{R} P^n$ called projective space?

I know that: If one defines an equivalence relation on $\mathbb{R}^{n+1}-\{0\}$ by $$x\sim y \iff y=tx$$ for some nonzero real number $t$, where $x,y\in\mathbb{R}^{n+1}-\{0\}$, Then The real ...
7
votes
2answers
143 views

$\mathbb{S}^2$ as a fibre bundle

I know, by the Hopf fibration, that $\mathbb{S}^3$ is an $\mathbb{S}^1$-fibre bundle over $\mathbb{S}^2$. Can $\mathbb{S}^2$ be an $\mathbb{S}^1$-fibre bundle over some manifold $M$?
3
votes
1answer
125 views

What's the point in Coordinate Functions?

A long time ago I've asked here about what O'Neill defines in his "Elementary Differential Geometry" book as "Natural Coordinate Functions". In the time, I've understood that it was a notational ...
6
votes
1answer
213 views

Difference between two definitions of Manifold

I've been studying Differential Geometry on Spivak's Differential Geometry book. Since Spivak just works with notions of metric spaces and analysis, I'm doing fine. The point is that Spivak presents ...
1
vote
0answers
104 views

Isomorphism of vector bundles (exercise 6.2 of Bott, Tu)

I'm self-studying the book by Bott & Tu "Differential forms in algebraic topology" and I'm having problems with exercise 6.2. It says "Show that two vector bundles on $M$ are isomorphic iff their ...
1
vote
0answers
22 views

Restriction to a line is an immersion.

I have proved that $g: \mathbb{R}^1 \rightarrow S^1, g(t) = (\cos 2 \pi t, \sin 2 \pi t)$, is a local diffeomorphism, as well as that $G: \mathbb{R}^2 \rightarrow S^1 \times S^1, G = g \times g$ ...
2
votes
1answer
116 views

Every diffeomorphism induces a bundle isomorphism?

I'm starting to learn some vector bundle theory and I have the next question. If I have a diffeomorphism $f:M \rightarrow M$ and $E$ is a vector bundle with base $M$, is it true that there exists a ...
0
votes
0answers
78 views

isomorphism of two compex line bundles

I am looking for some non-trivial examples of Line Bundles and an example about isomorphism of two line bundles. With details
2
votes
0answers
39 views

How are tangent spaces related?

Suppose we have a manifold with different smooth structures. The tangent space at a point depends on the choice of maximal atlas (right?). Is there a relation between the different tangent spaces? ...
7
votes
2answers
488 views

The Heisenberg manifold

I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a $3$-manifold which may be viewed as a circle bundle over ...
2
votes
2answers
38 views

Manifolds with finitely many ends

In the article ' The structure of stable minimal hypersurfaces in $ R^{n+1} $ ( http://arxiv.org/pdf/dg-ga/9709001.pdf) of Cao-Shen-Zhu the remark 2 at page 3 contains a statement that i don't ...
3
votes
1answer
73 views

Exact sequence arising from symplectic manifold

Let $M$ be a symplectic manifold, why is the following sequence exact? $$0\to \mathbb{R} \to C^\infty (M)\to A\to 0$$ Here $A$ is the set of global Hamiltonian vector fields.
-1
votes
1answer
449 views

definition of foliation in manifold and why foliation is useful? [closed]

I am thinking for the simple definition of "foliation" for a manifold. Why foliation is useful in manifold theory?
1
vote
1answer
90 views

integrability of ker $\omega$ in symplectic case

How can we prove that if $(M,\omega)$, is pre-symplectic and d$\omega=0$ then ker$\omega$ is integrable?.
2
votes
1answer
75 views

Hamiltonian reduction in symplectic geometry

If $V$ is symplectic and $W^\perp \subseteq W\subseteq V$, then why is $W$ a pre-symplectic vector space? Why is $W/W^\perp$ symplectic?
2
votes
2answers
70 views

Cutting the $2$-dimensional real Projective Space

I have the following question. Let $M$ be a smooth manifold which is homeomorphic to $\mathbb{R}P^{2}$. If one cuts $M$ along a non-contractible path then $M$ should be homeomorphic to a closed disc, ...
2
votes
2answers
107 views

Unfolding the $n$-dimensional sphere [duplicate]

Is there an extension to $n$ dimensions of the usual spherical coordinates mapping a three-dimensional sphere to a two-dimensional rectangle? [Duplicate]: Analogue of spherical coordinates in ...
3
votes
1answer
169 views

Laplace-Beltrami operator on sphere.

Suppose that we have solution of $$\delta d f = g$$ on sphere. Where $\delta d$ is Laplace-de Rham operator for functions, $f,g$ are scalar functions and $g$ has support on north hemisphere and it ...
4
votes
2answers
206 views

What is group manifold of a compact Lie Group?

I searched on google the meaning of a group manifold of a compact lie group but I didn't get the answer. A paper on arxiv "Background Independent Quantum Gravity:A Status Report- Abhay Ashtekar" on ...
4
votes
0answers
203 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ ...
1
vote
2answers
137 views

Smooth structure on the set of all straight lines

Consider the set of all straight lines on the euclidean plane $\mathbb R^2$. Introduce a structure of smooth manifold on this set and show that this is homeomorphic to $\mathbb R\mathbb P^2 ...
0
votes
1answer
68 views

Imbeddings of $n$-dimensional topological manifolds in $(2n + 1)$-euclidean space

H. Whitney proved that any $n$-dimensional smooth manifold $N$ can be imbedded in $(2n + 1)$-euclidean space (without any compactness assumption). If we consider the case of topological $n$-manifolds ...
1
vote
1answer
44 views

Question on the relationships of two and three manifolds

The Question is: Let $W_c = \{ ( x,y,z,w) \in R^4 | xyz = c \}$ and $Y_c = \{ ( x,y,z,w) \in R^4 | xzw = c \}$. For what real numbers $c$ is $Y_c$ a three-manifold? For what pairs $(c1,c2)$ is ...
1
vote
1answer
114 views

torus filling curve

I'm trying solve this problem but I didn't many ideas how to do it. So, if someone can give me a hint or the step of a solution I would greatly appreciate it. This is the problem: "Let ...
2
votes
1answer
85 views

Prove that the group $\langle U\rangle\leq G$ is open and closed in $G \subset GL_n\mathbb R$.

Let $G \subset GL_n\mathbb{R}$ be a closed subgroup and $U \subset G$ open with respect to the subspace topology. Prove that the group $\langle U \rangle$ generated by $U$, i.e. the smallest subgroup ...
4
votes
1answer
104 views

Theorem by Whitney

For $0<k<\infty$ and any $n$-dimensional $C^k$ manifold the maximal atlas contains a $C^\infty$ atlas on the same underlying set by a theorem due to Whitney. Could someone please point me to ...
1
vote
1answer
58 views

Integers or cantor set submanifold of the real numbers?

I'm trying to see whether $\mathbb{Z}$ or the cantor set $C$ are submanifolds or $\mathbb{R}$. Actually, I thought that $\mathbb{Z}$ was not a submanifold. As every subset of $\mathbb{Z}$ is ...
0
votes
1answer
301 views

Manifold being locally euclidean vesus Manifold being locally homeomorphic to an open set in $R^n$.

I was reading the definition of smooth manifold and i am little bit of confused. Informally it says A smooth manifold is a topological manifold (i.e. a topological space locally homeomorphic to a ...
2
votes
1answer
201 views

Importance of triangulation

Kervaire's seminal 1960 paper A manifold which does not admit any Differentiable Structure starts "An example of a triangulable closed manifold $M_0$ of dimension 10 will be constructed." What is the ...
0
votes
0answers
376 views

References for basic level Differentiable Manifolds and Lie Groups

I an undergraduate math student with a decent background in abstract algebra. I am looking forward to studying Lie groups this summer...I want some you to suggest good references for the following ...
2
votes
1answer
274 views

Smooth retraction onto a differentiable manifold

Let $M\subset\mathbb{R}^n$ be a smooth k-dimensional differentiable manifold (by which I mean that it is locally diffeomorphic to an open set in $\mathbb{R}^k$). Let us suppose $M$ compact for ...
16
votes
1answer
410 views

Is the fixed point set of an involution on a topological manifold a submanifold?

Let $f:X\to X$ be a homeomorphism of a topological manifold with $f^2=\mathrm{id}$. Is each connected component of $\{x\in X \mid f(x)=x\}$ a topological submanifold?
4
votes
2answers
91 views

Does $\mathrm{Mat}_{m \times n}$ have boundary?

To me, $\mathrm{Mat}_{m \times n}$ is isomorphic to $\mathbb{R}^{mn}$, hence is boundaryless. But this disqualified the use of Sard's theorem in this question: An exercise on Regular Value Theorem. ...
5
votes
1answer
70 views

Concerning the tangent space of an exotic $\mathbb R^4$

My geometric intuition is very poor, so my naive approach to this question is "if $M$ is an exotic $\mathbb R^4$, then $TM$ must be something like $\mathbb R^8$, which is not exotic". Of course, my ...