For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
2
votes
1answer
64 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 $I_2[Z_1,Z_2]=1$, ...
1
vote
0answers
40 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 ...
0
votes
0answers
30 views
Switching differentiation and integration on compact manifold
I'm looking for the theorem stating that differentiation and integration can be switched on compact manifold but I'm not sure there exists such theorem. Can anyone can state the theorem or tell me ...
3
votes
1answer
59 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
47 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
57 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
38 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
38 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
82 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 ...
0
votes
0answers
44 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
21 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
68 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 ...
3
votes
1answer
59 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
24 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 ...
2
votes
1answer
60 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?
6
votes
1answer
91 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
0answers
24 views
manifold projection
I found this paper:
Panoramic Mosaics by Manifold Projection
This paper mentions Manifold Projection. I understand its a combination of many projections.What exactly is it? What is the math behind ...
0
votes
1answer
23 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 $\psi_0=Id$ e $\psi_1=-Id$. Show this actions is properly discontinuos? The definition of ...
1
vote
0answers
50 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
70 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
27 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
108 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
149 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 ...
5
votes
1answer
72 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
60 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
70 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
47 views
How to calculate in local coordinates?
If $M$ is a smooth manifols what would be calculate a function $h$ defined on $M$ in local coordinates?
2
votes
0answers
35 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
57 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 ...
0
votes
2answers
78 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
47 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
53 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
91 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
votes
1answer
73 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
37 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
vote
1answer
24 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
vote
0answers
20 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 ...
0
votes
0answers
22 views
Closed subset of a manifold
I have a manifold M and a closed subset A on my manifold. When is a closed subset of a manifold a manifold?
1
vote
1answer
51 views
If $ \eta $ and $ \varphi $ are closed differential forms, then prove that $ \varphi \wedge \eta $ is a closed differential form.
Let’s assume that $ \eta $ and $ \varphi $ are closed differential forms. Then how can I prove that $ \varphi \wedge \eta $ is a closed differential form as well? Please explain how to solve this ...
1
vote
1answer
91 views
Prove that if $η$ is exact, then $η∧β$ is also exact.
Prove that if $η$ is exact, then $η∧β$ is also exact.
Please give a clear way to solve?
0
votes
1answer
81 views
how prove $\rho\wedge d\rho=0$ and how to show if $d(f\rho)=0$ for $f$ on $\Bbb R^{n}$ then $\rho\wedge d\rho=0$
$\def\d{\mathrm{d}}\def\R{\mathbb{R}}$Let $ρ$ be a $1$-form on $\R^{n}$ then
Firstly how to prove $\rho\wedge \d\rho=0$
Secondly, how to show that If $\d(f\rho)=0$ for some nowhere vanishing smooth ...
2
votes
1answer
51 views
Fundamental group of the following disc
What is the fundamental group of the following space in $\mathbf C^n$?
This is the topological space given by $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert ...
1
vote
0answers
17 views
paste Torus to itself
Suppose $Y$ is a topological space were obtained by pasting solid Torus to itself via the boundary map
$F:S^1\times S^1\to S^1\times S^1$, $F(z,w)=(z^aw^b,z^cw^d)$ where $a,b,c,d\in\mathbb{Z}$ and ...
2
votes
2answers
113 views
Normal coordinates
Let $M$ a riemannian manifold and $\nabla$ the Levi-Civita conection. Ineed to prove the next.
Let $B$ an open ball of radius $r$ in $T_pM$ such that $exp_p\mid _B$ be a difeomorphism over an open $U$ ...
0
votes
1answer
50 views
Smoothness Criterion for Vector Fields
I'm going to just write the proof (straight from Lee), my question is about the $(*)$ stared part.
Let $M$ be a smooth manifold and let $X:M \to TM$ be a (rough) vector field [all vector fields are ...
0
votes
0answers
42 views
Definition of g-orbit of a set
Let $g$ be a Lie algebra and $M$ a manifold, what does mean $g$-orbit of $M$?
0
votes
1answer
62 views
Submanifolds of Orientable Manifolds With Boundary
Let $(M, \partial M)$ be an orientable $n$-dimensional topological manifold with boundary. Suppose that $(N, \partial N)$ is an $n$-dimensional topological manifold with boundary and $N \subset M$.
...
1
vote
1answer
68 views
Show that the vector field $\operatorname{grad}f$ is smooth
Let $M$ be a Riemannian manifold and $ f:M\rightarrow\mathbb{R}$ be a smooth function. Define a vector field $\operatorname{grad}f$ in $M$ as
$$\langle\operatorname{grad}f,\,V\rangle=df(V)$$
for all ...
5
votes
1answer
233 views
Vector field and integral curve
Let M a riemaniann manifold, $V$ a vector field in M, and $\phi_t$ the respective flow
Let $p\in M$, and $\gamma$ the integral curve of $V$. Show that for all $v\in T_pM$
...
2
votes
1answer
102 views
Image of Homomorphism of Lie groups
This is exercise from Lee: Introduction to smooth manifolds.
Suppose $f \colon G \to H$ is homomorphism of Lie groups (real, finite-dimensional).
Q: Is image $Im(f) \subseteq H$ a Lie subgroup of H?
...
