0
votes
0answers
34 views

finite-dimensional continuous vector bundle

Let $M$ a compact metric space and $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric. I was wondering if it will be true that: ...
1
vote
1answer
26 views

Are vector bundles isomorphic when their transition functions are homotopic?

I'm in the process of trying to understand the equivalence between two different approaches to vector bundles, namely defining them to be smooth surjections $E\to B$ that are locally isomorphic to ...
-4
votes
1answer
269 views

Another vector bundles over of a Riemannian manifold

How is that any other vector bundle over a Riemannian manifold can be turn into a Riemannian manifold as well? I think that the question is of interest due to the presence in math.SE of several ...
0
votes
0answers
34 views

Finding frame bundles

Let $P\to M$ be a vector bundle of manifold $M$ with finite rank $n$. Is there any method to finding frame bundle $Fr(P)$ on $Fr(P)\to M$.
0
votes
0answers
9 views

Trivial Lie Algebroid

Let $A\longrightarrow M$ be a vector bundle, $\mathfrak{g}$ a lie algebra and $A:=TM\oplus (M\times \mathfrak{g})$. I read I can define a Lie bracket on the space of section $\Gamma(A)$ as follows, ...
2
votes
0answers
32 views

Restriction of a Lie bracket on the space of section of a vector bundle..

Let $A\longrightarrow M$ be a vector bundle and $U\subseteq M$ an open set. Suppose I have a lie bracket on $\Gamma(A)$ such that if $\rho:A\longrightarrow TM$ is a bundle map then $$[a, fb]=f[a, ...
0
votes
1answer
23 views

What would be the space of section of the bundle $\mathfrak{g}\longrightarrow \{e\}$?

Let $\mathfrak{g}$ be a Lie algebra and $\pi:\mathfrak{g}\longrightarrow \{e\}$a vector bundle over a point. What would be the sections of this bundle?
1
vote
1answer
41 views

a question about compact tangent bundle

I have a question about tangent bundles. Is there a compact tangent bundle? Or what conditions do we need to be sure that tangent bundle of a manifold be compact?
2
votes
2answers
39 views

Tangent bundle of sphere with $g$ handles

How can one show that tangent bundle $TM$ is not trivial if $M$ is a sphere with $g$ handles and $g \ne 1$?
1
vote
1answer
83 views

How to prove that a vector bundle is trivial iff there are n global sections that form a basis on each fiber?

I can prove the only if part. My attempt to prove if part is the following: Given $n$ global sections $s_1, s_2, ..., s_n$ of a vector bundle $E$ on a smooth manifold $M$ such that they form a basis ...
0
votes
1answer
57 views

Cotangent bundle of n-dimensional diferentiable manifold is 2n-dimensional manifold

How to prove that cotangent bundle of n-dimensional diferentiable manifold is 2n-dimensional manifold? Detailed explanation is welcome. Thanks in advance.
0
votes
0answers
50 views

Constructing vector bundles from local covers and transitions functions

Let $M$ be a smooth manifold. Suppose we are given an open cover ${U_\alpha}$ of $M$ and for $\alpha,\beta$ ; a smooth map $\tau_{\alpha\beta}\colon U_\alpha \cap U_\beta \to GL(k; R)$ satisfying the ...
0
votes
0answers
47 views

Sections on the Tautological Line bundle $E(\gamma_n)$..

I have a question about the tautological line bundle over $\mathbb R\mathbb P^n$. Recall, this bundle is that whose total space is $$E(\gamma_n):=\{([x], v)\in\mathbb R\mathbb P^n\times \mathbb ...
2
votes
2answers
54 views

Problem with normal bundle to sphere $\mathbb S^n\subset \mathbb R^{n+1}$

I'm in trouble for understanding the normal bundle to $\mathbb S^n\subset\mathbb R^{n+1}$. By definition $$\nu(\mathbb S^n)=\bigcup_{p\in\mathbb S^n}\nu_p(\mathbb S^n),$$ where $\nu_p(\mathbb ...
1
vote
1answer
83 views

Normal Bundle is Trivial

how can I show the normal bundle $$\nu(\mathbb S^n):=\bigcup_{p\in\mathbb S^n} T_p\mathbb R^{n+1}/T_p\mathbb S^n, $$ is a trivial bundle? Any help will be valuable.. Thanks
4
votes
1answer
53 views

How to show $f^*E$ is a smooth submanifold…

I'm wondering how to show the following: let $E$, $B_1$ and $B_2$ smooth manifolds. Suppose $\rho:E\rightarrow B_2$ is a smooth vector bundle and $f:B_1\rightarrow B_2$ a smooth map. If we write ...
6
votes
1answer
148 views

What's the intuition behind the tangent bundle?

Well, when we work with a smooth manifold $M$ we can associate with each point $p\in M$ a vector space $T_p M$ of all vectors at $p$ tangent to $M$: this is the space of linear functionals obeying ...
4
votes
1answer
141 views

Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
0
votes
1answer
121 views

Understanding the definition and meaning of cotangent space

I would like to understand definition and also meaning of cotangent space. Let's first of all define tangent space: generally we know that equation of tangent line of function $f(x)$ at point $x_0$ ...
1
vote
0answers
57 views

Tangent bundle of a quotient manifold

I am interested in the tangent bundle of the quotient of a manifold by a proper and free action but I can't find any reference on the net. Does anyone know a book or article where it is described ?
1
vote
0answers
57 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 ...
2
votes
1answer
61 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
42 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
3
votes
0answers
46 views

Maps between total spaces of holomorphic vector bundles

I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles. Let me outline a situation that is a bit more concrete, to help focus ...
4
votes
1answer
214 views

Classifying vector bundles

Given a manifold $M$, is there a way of classifying up to isomorphism all possible vector bundles over $M$ of a given rank? Some other questions on this site deal with specific cases, which all seem ...
1
vote
0answers
60 views

Bott connection

Can anyone help me showing the following: Let $E$ be a smooth vector bundle over $M$ and $F\subseteq TM$ a smooth distribution. A $F$-connection is a $\mathbb R$-bilinear aplication ...
1
vote
1answer
48 views

Distribution and Tangent Bundle

Let $F=\{F_p; p\in M\}\subseteq TM$ be a rank $k$ smooth distribution. Can anyone explain-me what is the set $$\displaystyle\nu(F)=\frac{T_pM}{F_p}.$$
4
votes
1answer
98 views

Vector Bundle Doubt..

Well I have a doubt about a rank $k$ vector bundle. My definition of vector bundle is: A rank $k$ vector bundle is a triple $(\pi, E, M)$ where $E$ and $M$ are smooth manifolds and $\pi:E\rightarrow ...
3
votes
1answer
136 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?
7
votes
1answer
143 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 ...
9
votes
1answer
253 views

Vector Bundle Over Contractible Manifold

The problem comes from Liviu Nicolaescu's book Lectures on the Geometry of Manifolds. He asks the reader to prove that any vector bundle $E$ over $\mathbb{R}^n$ is trivializable. The idea he gives is ...
3
votes
1answer
100 views

Any vector bundle on $\mathbb R$ is a trivial bundle

How to prove that any vector bundle on a Euclidean space is a trivial bundle? It is enough to prove it for the case of dimension $1$ and I hope it will be a nice exercise for me to generalize to the ...
4
votes
1answer
111 views

Tangent bundle : is a manifold

I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector ...
3
votes
1answer
241 views

Trivial tangent bundle and orientability

Let $M$ a (real) $n$-dimensional connected differentiable manifold. (a) The tangent bundle $TM$ is trivial, $TM \simeq M \times \mathbb R^n$; (b) $M$ is orientable. Consider the ...
4
votes
2answers
206 views

Describe tangent and normal bundle to a manifold

Consider the set $X:=\{(x,y,z)\in\mathbb{R}^3 | x^2+3y^2=1+z^2\}$ I have to show that $X$ is a submanifold of $\mathbb{R}^3$ (and this is trivial); then, using on $T\mathbb{R}^3$ the standard ...
8
votes
1answer
266 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 ...
2
votes
1answer
81 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
0answers
57 views

Second order equations on manifolds

In my notes, the lecturer considers a smooth vector field $v: TM\to T(TM)$, with $M$ a smooth manifold. Let's write $$v(u,e)=((u,e), (a(u,e),b(u,e)).$$ It is said that $v$ is a second order equation ...
1
vote
2answers
272 views

Extending a Set of Linearly Independent Vector Fields to a Basis

My question is this. Suppose we are given some smooth vector fields $X_1, X_2,..., X_k$ which are linearly independent at all points in a neighborhood $U$ (EDIT: diffeomorphic to a ball) of $R^n$. Do ...
3
votes
2answers
331 views

Any example of manifold without global trivialization of tangent bundle

It is said for most manifolds, there does not exist a global trivialization of the tangent bundle. I am not quite clear about it. The tangent bundle is defined as $$TM=\bigsqcup_{p\in M}T_PM$$ So is ...
0
votes
0answers
36 views

A good way to embed a manifold in a Euclidian space $\mathbb{R}^n$

We know that any closed manifold $X$ can be embedded into some Euclidian space $\mathbb{R}^n$ for sufficiently large $n\in \mathbb{N}$. What is the easiest way to see this fact? I have seen several ...
4
votes
0answers
153 views

de Rham Cohomology of Non-Flat Bundle

Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$. If $E$ ...
3
votes
1answer
158 views

Trivialisation of the normal bundle of $S^1$

I'd like to show that the normal bundle of $S^1$ is trivial. That is, I want to find a homeomorphism $\varphi : NS^1 \to S^1 \times \mathbb R$ such that $\varphi \mid_{N_s S^1}$ is linear for every $s ...
5
votes
0answers
113 views

Alternate pullback bundle construction

If $\pi : F \to N$ is a fiber bundle, and $\phi : M \to N$ (here, $M$ and $N$ are manifolds), then the standard way to define the pullback bundle of $F$ by $\phi$ is $$\phi^* F := \{(m,f) \in M ...
2
votes
1answer
209 views

Sum of normal bundle and tangent bundle.

Would somebody be able to prove that the whitney sum of the normal and tangent bundles of a submanifold of R^n is trivial? Would apreciate a detailed proof...I'm struggling a little. Tina
1
vote
0answers
376 views

Trivial tangent bundle and parallelizability of a $n$-sphere

So, I want to show that a $n$-sphere $S$ is parallelizable iff it has trivial tangent bundle. For "$\Leftarrow$" I would like to take a trivialization $\varphi:S\times \mathbb{R}^n\rightarrow TS$ and ...