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

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2
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1answer
339 views

Tangent space to a product

Can you explain this question explicitly. This is a little bit difficult for me, but I want to learn how to solve. Thank you for help. If $M$ and $N$ are manifolds, let $\pi_1:M\times N\to M$ and ...
1
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2answers
67 views

Compute the velocity vector.

Can you solve explicitly? please. I don't know how to solve. Thank you for help.
-1
votes
1answer
64 views

Is $S$ a regular submanifold?

$M=M_{n\times n}(\Bbb R)$ $S=\operatorname{SL}(n, \Bbb R) = \left \{ A \in M \mid \det(A)=1 \right \}$ $M$ is an $n^{2}$ dimensional $C^{\infty}$ manifold. Is $S$ a regular ...
1
vote
1answer
337 views

Determining push forward of a vector field of a submanifold

Let $M,N$ be two differentiable manifolds, $\phi:M\to N$ a diffeomorphism and $X$ a vector field on $M$. For example, one can determine the push forward ...
0
votes
1answer
59 views

Is $S$ a regular submanifold of $\Bbb R^{3}$?

$$S=\{(x,y,z) \mid x^{2}+y^{2}=z^{2}\}$$ $g: \Bbb R^{3}\to \Bbb R$, $S=g^{-1}(0)$ Is $S$ a regular submanifold of $\Bbb R^{3}$? I'd be grateful for a clear and explicit explanation of why this is ...
5
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2answers
168 views

Differentiable manifolds $\mathscr C^k$ vs. $\mathscr C^\infty$

I noticed that there exists a (in some sense) better definition of the tangent space via the dual of a certain quotient algebra which is easier to work with in some cases. This however only works for ...
2
votes
0answers
63 views

Torus biholomorphic to smooth cubic curve?

I am trying to understand that all compact genus 1 Riemann surfaces are biholomorphic to a smooth cubic curve. ( assuming that $\dim H^{1,0} = \dim H^{0,1} = \frac{1}{2} \dim H^1$ ) I think I ...
1
vote
1answer
85 views

Difference between “Live” and “Define”

In many mathematical text to determine an object on manifold, the verbs "live" and "define" are used. I'm interested to know whether there is a difference between the concepts of "to define" and "to ...
3
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1answer
106 views

Approach topological manifolds with smooth manifolds

Because I'm doing some problems that consider all the manifolds while the situation is really clear when considering only smooth manifolds. Thus my question is can we always appoint a topological ...
6
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1answer
206 views

Prove Poincare duality theorem with Morse theory.

First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
1
vote
1answer
132 views

Structure on manifolds

Thank you for your attention. First, I would like to know why we see some different structures defined on manifolds: What is the necessity to have different structures, like Kähler structure, ...
0
votes
1answer
69 views

Metric Spaces needed for Differential Geometry

I've asked here about some texts about differential geometry which doesn't assumes that the reader knows general topology. I've got good references as Do Carmo's Differential Geometry of Curves and ...
1
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0answers
118 views

About the Morse theory

I am trying to study the Morse theory and would like to know the purpose of this study, why when we talk about the critical point of a manifold is mentioned Morse theory? , are the critical points of ...
1
vote
1answer
87 views

Doubt about the Domain of the chart on a Manifold

I have a doubt about the domain of the chart on a manifold. Suppose $M$ is a smooth manifold and that $(U, \varphi)$ is a chart on $M$, then $\varphi : U \to \mathbb{R}^n$ has $U$ as it's domain. ...
0
votes
1answer
121 views

Problem related to differential of a map

I dont understand how to solve this problem. Please can you explain the solution clearly? I want to learn how to solve such problems. Thank you
2
votes
1answer
93 views

Existence of a map homotopic to an odd degree map which is transversal to transversal submanifolds?

my question is as follows: Given a map $f:X\rightarrow Y$ between compact spaces of equivalent dimension, and two sub-manifolds $Z_1,Z_2 \subset Y$ which transversely intersect with $I_2[Z_1,Z_2]=1$, ...
2
votes
0answers
224 views

Flow of a complex vector field?

Suppose I have a vector field X over a m-dimensional analytical manifold $M\subset \mathbb{C}^n$; how can I define the flow of $X$? Is it done in the same way as for the real case, but instead of ...
5
votes
1answer
300 views

Hopf Fibration in Local Coordinates

I have the following task: Consider the unit sphere $\mathbb S^3$ in $\mathbb R^4$. We know $\mathbb CP^1\simeq \mathbb S^2$ (homeomorphic). Identifying $\mathbb R^4$ with $\mathbb C^2$, we have a ...
1
vote
1answer
250 views

Tangent bundle on a complex manifold

When defining the tangent bundle of a $n$-dimensional manifold $M$ whose coordinate-change mappings are holomorphic, do I need to specify where M is immersed? That is, I must assume that ...
1
vote
1answer
142 views

Differentiable manifold in dimension 1 and its critical point

Please, I want to know how to define a differentiable manifold in dimension 1, and if the circle is a differentiable manifold in dimension $1$, and what is its critical point. Thank you.
1
vote
1answer
43 views

Is $VV^T + D$ a submanifold?

If the positive definite matrix P forms a manifold, is that the subset that {P: P = V V^T + D} where V is a low rank matrix and D is a positive definite matrix a sub-manifold? This idea is ...
1
vote
0answers
81 views

Smoothing corners of a handle attachment

Say we attach a $\lambda$-handle, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, to a smooth manifold $M, \partial M$ by simply taking the quotient $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ ...
7
votes
2answers
138 views

Finding a subspace whose intersections with other subpaces are trivial.

On p.24 of the John M. Lee's Introduction to Smooth Manifolds (2nd ed.), he constructs the smooth structure of the Grassmannian. And when he tries to show Hausdorff condition, he says that for any 2 ...
1
vote
1answer
267 views

How general formulation of Stoke's theorem relate to Kelvin-Stokes theorem

I asked a similar question, but I realized the question is too vague and it's better to start a new one: We know that there are two usually used formulations of Stoke's theorem. One is vector ...
2
votes
0answers
34 views

Exponential Families and Riemannian Symmetric Spaces

Suppose the $f_{X}(x|\theta)$ is a probability density function from an exponential family. Is it true that the Riemannian manifold which has the Fisher information as it's Riemannian metric is a ...
2
votes
2answers
523 views

Compact manifolds and orientability

I've a doubt about compact manifolds and orientability. I know that Compact Manifolds in $\mathbb{R^3}$ are orientable. My questions is: The statement above is valid only for compact manifolds ...
4
votes
1answer
139 views

On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where ...
1
vote
1answer
58 views

Question about Interior product computation

I need to evaluate $\omega=i_X(dx\wedge dy)$ where $X$ is a vector field in $\mathbb{C}^2$ (which means $p=2)$. If I write $X(x,y)=(X_1(x,y),X_2(x,y))$, or simply $X=(X_1, X_2)$, then the interior ...
4
votes
1answer
225 views

Trivial Tangent and Cotangent Bundles

If we have a smooth manifold $M$, why is the tangent bundle $TM$ trivial (as a vector bundle) iff the cotangent bundle $T^*M$ is trivial as well?
10
votes
2answers
213 views

Ways to think about vector bundle

I'm studying manifold theory and I've got to the point of discussing the definition of a vector bundle. The definition is quite long and a bit confusing and I was wondering if someone with a bit more ...
0
votes
1answer
54 views

Prove this action is properly discontinuous..

Consider the group $\mathbb Z_2=\{0, 1\}$ acting on the sphere $\mathbb S^n$ through the group actions $\psi_0=Id$ and $\psi_1=-Id$. Show this actions is properly discontinuos? The definition of ...
4
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0answers
328 views

Differentiable Manifold Hausdorff, second countable

Why do we generally require that a differentiable manifold be Hausdorff and second countable? Is this universally accepted in the definition? My Professor only required the Hausdorff condition for ...
2
votes
2answers
117 views

How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$?

Suppose the additive group $\mathbb Z^n$ acts on $\mathbb R^n$ through translation. How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$? The translation action is given by ...
1
vote
1answer
50 views

How to show the orbit space $\mathbb S^n/\mathbb Z_2$ is $\mathbb RP^n$?

How to show the orbit space $\mathbb S^n/\mathbb Z_2$ is $\mathbb RP^n$? Here $\mathbb Z_2=\{0, 1\}$ is the additive group and the group action considered induces the aplications $\psi_0=Id$ and ...
4
votes
1answer
326 views

Why do we need Lie derivative?

If a manifold is equipped with Levi-Civita connection, Why do we need Lie derivative? In Euclidean space to calculate directional derivative of a vector field V along W, we parallel transport V along ...
4
votes
1answer
261 views

Different definitions of handle attachment

This is an extremely technical question about handle attachments.  I am asking why two definitions are equivalent.  My question appears in the second to last paragraph after I've described the two ...
4
votes
1answer
259 views

A question about concept of pushforward

In An Introduction to Smooth manifolds by Lee is written: for any smooth vector fields V and W on a manifold $M$, let $\theta$ be the flow of $V$, and define a vector $(\mathcal{L}_v W)_p$ at each ...
4
votes
2answers
196 views

Closed ball not a manifold.

My book on differential geometry claims that a closed ball in $\Bbb R^n$ can never be a differentiable manifold because of the boundary points. The book doesn't really give an explanation for why this ...
2
votes
0answers
108 views

Vanishing of local cohomology of constructible sheaves

Recall, that if $\mathcal{F}$ is a coherent sheaf on a variety and $Z$ is an l.c.i. subvarity of codimension $n$, then $H^i_Z(\mathcal F)$ vanishes for $i > n$. Is there an analogous statement for ...
-1
votes
1answer
121 views

How to calculate in local coordinates? [closed]

If $M$ is a smooth manifols what would be calculate a function $h$ defined on $M$ in local coordinates?
3
votes
0answers
78 views

Show that $\Gamma(TM) \cong_{\mathbb{R}} \mathsf{Der}_{\mathbb{R}} C^{\infty}(M)$

Let $M$ be a smooth real manifold. I want to show that we have an isomorphism of real vector space $\Gamma(TM)$ of all smooth sections of $TM$ (i.e. of vector fields on $M$) and of real vector space ...
3
votes
2answers
75 views

Homeomorphism between $\mathbb{R}_+ \times \mathbb{R}_+$ and $\mathbb{R}_+ \times \mathbb{R}$

I am trying to find a (smooth, if possible) homeomorphism between $\mathbb{R}_+ \times \mathbb{R}_+$ and $\mathbb{R}_+ \times \mathbb{R}$. i have come with some ideas, but the resulting functions are ...
1
vote
2answers
551 views

The real projective space $RP^{n}$ is second countable.

. The real projective space $RP^{n}$ is second countable. How to prove this. I have to use this proof in a solution of a question. But I cannot prove. Please help me. Write the proof clearly.
0
votes
1answer
116 views

Prove that a surface of revolution is a 2dimension manifold

I have a question about surface of revolution. Prove that a surface of revolution is a 2dimension manifold.
5
votes
1answer
102 views

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
7
votes
2answers
238 views

Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
2
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1answer
201 views

prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.

A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ R_{n}$ and $V ⊂ R_{m}$ are homeomorphic, then n = m. prove that the sphere with a ...
0
votes
1answer
49 views

Show that $af∧bg=(ab)f∧g$

Let $v$ be vector space. For $a$ and $b$ are in IR, $f$ is in $A_{k}(V)$ and $g$ is in $A_{l}(V)$ Show that $af∧bf=(ab)f∧g$ Here Will I use the definition of wedge product? Is ti right? How to use? ...
1
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1answer
53 views

set of immersions on a manifold is an open set in the set of all mappings to $\mathbb{R}^{2m+1}$

I'm studying a proof of Takens' Delay Embedding Theorem. A key fact used is that the set of immersions is open in the set of all mappings (mappings from an $m$-dimensional manifold M to ...
1
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0answers
38 views

Coefficients Relative to a Smooth Frame

An exercise from Loring Tu's textbook asks the following question: Let $\pi:E\to M$ be a $C^\infty$ vector bundle and $s_1,\ldots,s_r$ a $C^\infty$ frame for $E$ over an open set $U$ in $M$. Then ...