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

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4
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54 views

non-constant curve c with $\frac{\mathrm{d}c}{\mathrm{d}t}= \lambda\frac{\mathrm{d}^2c}{\mathrm{d}t^2}$

I don't know how to solve the following problem and would appreciate some help. Let $M$ be a submanifold of euclidean space and $c:[a,b] \to M$ a non-constant curve, such that the velocity field of ...
0
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1answer
142 views

How can I prove that “If $M$ is contractible differentiable manifold, then $M$ is orientable?”

If $M$ is a contractible differentiable manifold, then $M$ is orientable.
2
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1answer
83 views

Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations

Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$ Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on ...
1
vote
1answer
79 views

Submanifolds of a space of functions?

Let $f:\mathbb{R}\rightarrow(\mathbb{R}\rightarrow\mathbb{R})$ be a function mapping a real number uniquely into the set $\mathbb{F}$ of total functions from $\mathbb{R}$ to $\mathbb{R}$. $\mathbb{F}$ ...
2
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1answer
108 views

a theorem in topology

Is there anyone know there is a theorem in topology which states that a compact manifold "parallelizable" with N smooth independent vector fields. must be an N-torus? and why the vector field here is ...
4
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1answer
107 views

$U(n)/U(n-1)$ as homogeneous space

How can I prove that the quotient $U(n)/U(n-1) \simeq S^{2n+1}$ (where $U(n)$ is the unitary group). Is il correlated with the teory of homogeneous spaces?
3
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1answer
73 views

Is there any manifold that is not a subspace of a finite dimensional euclidean space?

I mean, is there any topological space that is locally euclidean, Haudorff and second countable and can't be embedded into a finite dimensional Euclidean space. I think it's hard for me to find such ...
13
votes
1answer
543 views

Showing $[0,1] \times [0,1]$ is a manifold with boundary

I'm familiarizing myself with manifolds. I tried to show $[0,1]\times[0,1]$ is a manifold with a boundary. Can you please tell me if my proof is correct: The definition for manifold with boundary: ...
2
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1answer
219 views

Why is this not a proof of Invariance of Domain?

We know that if $f:K \to X$ is continuous and injective, $K$ is compact, and $X$ is Hausdorff, then $f$ is a homeomorphism $K \cong f(K)$. So suppose $f:U \to \mathbb{R}^n$ is continuous and ...
1
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0answers
66 views

Velocity vectors on $S^{3}$

Consider $S^{3}$ as the unit sphere in $C^{2}$ under the usual identification $C^{2}\leftrightarrow R^{4}$. For each $z=(z^{1},z^{2})\in S^{3}$, define a curve $\gamma _{z}:R\rightarrow S^{3}$ by ...
1
vote
1answer
51 views

Question on Notation

In Loring Tu's text An Introduction to Manifolds, Exercise $2.4$ asks us to show that $D_1\circ D_2$ need not be a derivation while $D_1\circ D_2-D_2\circ D_1$ is always a derivation. My question is ...
0
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0answers
91 views

Covering space (Lie groups and their maximal tori)

Let $ G $ be a compact Lie group and $ T $ a maximal torus in $ G $. We define the Weyl group $ W $ as the quotient space $ {N_{G}}(T)/T $, where $ {N_{G}}(T) $ is the normalizer of $ T $ in $ G $. We ...
2
votes
2answers
146 views

The most general notion of a directional derivative

Questions: I know you can define a directional derivative on some subset of $\mathbb R^n$, but what can be said about an arbitrary set of points, $S$? What are the most general criteria $S$ must ...
1
vote
2answers
152 views

The dimension of linear map

I am reading "Introduction to smooth manifolds" by Lee and one place is very unclear for me: Let $P$ and $Q$ be any complementary subspaces of $V$ (which is an $n$-dimensional real vector space) of ...
2
votes
2answers
121 views

Gluing cylinders together

I was trying to glue two cylinders and then show that the resulting space is a manifold. Here is the first of my attempts: The cylinders I denote $C_0 = S^1 \times [0,1]$ and $C_1 = S^1 \times ...
5
votes
0answers
120 views

Gluing manifolds

Is it true that gluing can be realized two ways? In the example of gluing two cylinders to obtain a torus one can imbed each cylinders into the torus like this: A point $(\cos \phi , \sin \phi, ...
2
votes
1answer
71 views

Neighborhoods of half plane

Define $H^n = \{(x_1, \dots, x_n)\in \mathbb R^n : x_n \ge 0\}$, $\partial H^n = \{(x_1, \dots, x_{n-1},0) : x_i \in \mathbb R\}$. $\partial H^n$ is a manifold of dimension $n-1$: As a subspace of ...
1
vote
1answer
149 views

What is the characteristic property of surjective submersions?

In Lee's 'Introduction to smooth manifolds' he states that given smooth manifolds $X,Y$ and a surjective submersion $f:X\to Y$, then $f$ is a smoothly final map, that is for any further smooth ...
2
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0answers
68 views

Hodge decomposition on a manifold with a nontrivial connection

I am familiar with the notion of Hodge decomposition of an arbitrary differential form into an exact form, a co-exact form, and a harmonic form. Given a curved space with a connection, could you ...
1
vote
1answer
248 views

any two simply connected open set in the plane R^2 are diffeomorphic

Prove that any two simply connected open set in the plane R^2 are diffeomorphic. I know that in the complex plane any simply connected open set is diffeomorphic to either complex plane or open unit ...
13
votes
1answer
323 views

Manifold of Density Matrices

Let $\mathrm{M}_{d\times d}\left(\mathbb{C}\right)$ denote the set of all $d\times d$-matrices with complex entries. My goal is to show that the set $\mathcal{M}:= \left\{ \rho\in \mathrm{M}_{d\times ...
2
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0answers
99 views

Can I flip orientation at a point of a non-orientable manifold?

Let $p \in M$ be a point of a non-orientable smooth manifold, $M$. Does there exist a diffeomorphism $f: M \rightarrow M$ with $p \mapsto p$ and such that $df : T_pM \rightarrow T_pM$ is orientation ...
1
vote
0answers
75 views

transformation of symplectic structure by a matrix

Suppose that in canonical symplectic basis $e_1,e_2,f_1,f_2$ we have $$\Omega=pf_1^*\wedge f_2^*+qe_1^*\wedge e_2^*+r(e_1^*\wedge f_2^*+e_2^*\wedge f_1^*)+s(e_1^*\wedge f_1^*-e_2^*\wedge f_2^*)$$ Let ...
2
votes
1answer
200 views

proof of preimage theorem for abstract manifolds?

How is the preimage theorem proved for abstract manifolds (not necessarily embedded in $\mathbb{R}^N$)? It states that if $F:M\rightarrow N$ is a smooth map between smooth manifolds and $q\in N$ and ...
3
votes
2answers
604 views

Pullback of differential form of degree 1

Good evening, In differential forms (in the proof of the naturality of the exterior derivative), I don't get why if $h\in \Lambda^0(U)$ and $f^*$ is the pullback then, $f^*dh=d(f^*h)$. I wrote ...
2
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0answers
211 views

Use of Implicit Function Theorem to provide examples of Manifolds

Suppose we know Definition 1 and Theorem 1, and want to prove Theorem 2 given below. Definition 1: A subset $M$ of $R^n$ is called an k-dimensional manifold (in $R^n$) if for each point $x\in M$ the ...
11
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1answer
355 views

Second Stiefel-Whitney Class of a 3 Manifold

This is exercise 12.4 in Characteristic Classes by Milnor and Stasheff. The essential content of the exercise is to show that $w_2(TM)=0$, where $M$ is a closed, oriented 3-manifold, $TM$ its tangent ...
1
vote
1answer
155 views

equivalence of different definitions of isotopy

Here are two supposedly equivalent definitions of a smooth isotopy (M and N are smooth manifolds): A smooth level preserving imbedding $M \times I \rightarrow N \times I$ A smooth map $ F: M\times I ...
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0answers
36 views

Example of atlas for sequence space

Is it possible to construct and atlas for space $\ell_1$? I.e. give an example of collection $(U_\alpha,\phi_\alpha)$, such that $\cup U_\alpha=\ell_1$ and $\phi_\alpha:U_\alpha\to R^{n}$ is a ...
2
votes
1answer
107 views

hairy ball thm. and projective space

Is it possible to find $n>1$ such that $\mathbb{R}P^{2n+1}$ doesn't have smooth non vanishing vector field? I know it is not true for $S^{2n+1}$ and $\mathbb{R}P^{2n+1}$ is a sphere modolu antipod ...
2
votes
1answer
301 views

Manifolds are paracompact

Every manifold is paracompact. I tried: $M$ is an $n$--manifold with open covering $U_\alpha$ and $\varphi_\alpha$ local homeomorphisms; $\varphi_\alpha (U_\alpha)$ are open in $\mathbb R^n$. Adding ...
3
votes
2answers
228 views

Learning about Manifolds

I am looking to learn about manifolds for use in signal processing. I have a engineering degree where I have covered calculus and basic linear algebra, with this background in mind, does anyone have a ...
2
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0answers
117 views

To what extent is the global angular form well-defined?

I am reading Differential Forms in Algebraic Topology by Bott & Tu. The authors constructed the global angular form $\psi$ for an oriented $k$-sphere bundle $E$ over a smooth manifold $M$. It has ...
1
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2answers
48 views

Find $A^{-1}$(W) of linear manifold W

Given linear map $A:\mathbb{R}^2\to \mathbb{R}^4$ defined as $$A = \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ 0 & 2 \\ 3 & 1 \end{pmatrix}$$ and linear manifold $ W \subset ...
9
votes
1answer
176 views

When does the quotient of a manifold with boundary become a manifold?

Given a manifold $M$ with boundary $\partial M \neq \varnothing$, when can we form a manifold $\tilde M$ from $M$ by collapsing the boundary? In the examples I've considered it seems like collapsing ...
2
votes
1answer
85 views

Open sets of the tangent bundle in a Riemannian manifold

Let $M$ be a Riemannian manifold with a metric $g$ and $(U,\varphi)$ a chart around a point $p\in M$. By a Remark page 63 of Riemannian Geometry by M. Do Carmo, it seems that any open set ...
3
votes
1answer
174 views

Difference between tensor and tensor field?

I couldn't get the difference between tensor and tensor field. I'm learning from Barret O'neill's Semi-Riemann Geometry and here are the definitions: if $A:(V^*)^r \times V^s\to K$ transformation is ...
1
vote
1answer
253 views

Manifolds and level sets

Let $M$ be the set of points $(x,y,z)$ in $\mathbb{R}^3$ such that $x^2+y^2+z^2=1$ and $x^2=yz^2$. The point $(0,-1,0)$ is removed. The question is: after removing a second point (to determine), why ...
1
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2answers
514 views

Flat manifold and metric

Is "X is a flat Riemannian manifold" equivalent to "for any metric g on X, there is a change of coordinates that can transform g to a tensor with only constants on its diagonal"?
2
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1answer
62 views

Properties of $S_2$ and the plane and $[−1,1]^2$

The question: Is the sphere $S_2$ isometric / diffeomorphic / homeomorphic to the plane? Is the sphere $S_2$ minus a point isometric / diffeomorphic / homeomorphic to the plane? Is the sphere $S_2$ ...
0
votes
1answer
47 views

the tangent bundel TM of a manifold is also manifold of dimension twice the dimension of M

I want to prove that the tangent bundel TM of a manifold is also manifold of dimension twice the dimension of M? could you help me? Thanks!
3
votes
1answer
206 views

Submersion Theorem for Banach Spaces

I'm having difficulty proving a well-known result from functional analysis. Any hints would be greatly appreciated. Fix a Fréchet differentiable map of Banach spaces $g: X \to B$. Assume that, at a ...
2
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0answers
95 views

Partition of Unity for the Divisor Sheaf

Recall that given a Riemann Surface $X$, the divisor sheaf is the sheaf ${\cal D}$ which assigns to each open set $U$ the collection of maps $\phi:U \to \mathbb{Z}$ such that $\phi(p)=0$ for all but ...
14
votes
1answer
188 views

Are locally homotopic functions homotopic?

Suppose we have two (smooth) functions $f,g:X\to Y$, where $X,Y$ are smooth (second-countable, Hausdorff) manifolds which are locally homotopic (that is, any point in $X$ has a neighbourhood $U$ such ...
26
votes
2answers
811 views

PDEs on manifold: what changes from Euclidean case?

I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds. For example, do things like Poincare's inequality ...
3
votes
1answer
152 views

Coordinate representation for given tangent space

Let $M$ be differentiable n-manifold. Suppose that at $p \in M$ we are given a basis of tangent space $T_pM$ denoted as $(X^1,\ldots,X^n)$. Can we construct a coordinate representation ...
5
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0answers
160 views

Does every non-compact manifold admit an incomplete vector field?

I know that every vector field on a compact manifold is complete. The question of whether every non-compact manifold admits an incomplete vector field seems to follow naturally. I'd hazard a guess ...
1
vote
0answers
123 views

Is every manifold sigma-compact?

Thanks for reading my post. Here is my question. I want to know if every surface is hemicompact, i.e., there is a compact exhaustion. I think that question could be asked for every manifold. I know ...
10
votes
1answer
367 views

uniqueness of the smooth structure on a manifold obtained by gluing

I've just read a proof that If $M$, $N$ are smooth manifolds with boundary and $f: \partial M\rightarrow \partial N$ is a diffeomorphism then $M \cup_f N$ has a smooth manifold structure such that ...
2
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0answers
246 views

Orientation of manifold in topological sense

What do we mean by orientability of a topological manifold? How do we orient two dimensional Euclidean space and why is Moebius band non-orientable? And it would be a great favour to me if you can ...