1
vote
1answer
25 views

spin structure definition

Suppose we have a principal $SO(n)$-bundle $E$ over $B$, with projection map $p$. We say that it admits a spin structure if there is a prinicipal $spin(n)$-bundle $E'$ over B, with projection map ...
0
votes
0answers
26 views

Sections of endomorphisms of a vector bundle

The following could be rather silly question but I haven'd found it stated explicitly; from the other side, it seems to me, that this fact is used often without comments. The problem is the ...
0
votes
0answers
18 views

Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
1
vote
1answer
54 views

Group of Automorphism of the Fiber of Fiber Bundle

Let us consider the Mobius Bundle. Can Someone Explain the part "There is no natural unique homeomorphism of $Y_x$with $Y$. However there are two such which differ by the map $g$ of $Y$ on itself ...
1
vote
1answer
28 views

Well-definedness of the action of the structure group of a principal bundle on the total space.

Find the definition of a fiber bundle here- Definition of Fiber Bundle I am having difficulty in proving that the natural action of $K$ on $X$ is well-defined: Let us recall how does K acts on X ...
0
votes
1answer
16 views

Linear clutching function properties

I'm trying to prove proposition $4.8$ of Husemöller (pay 148) Here it is the text: Where the only hints are: $[L^{n+1}(\xi),L^{n+1}(zp)] \approx [L^{n}(\xi),L^{n}(p)] \oplus [\xi,z]$ ...
4
votes
1answer
41 views

Characteristic classes for quaternionic bundles

In a nutshell, the derivation of the Stiefel-Whitney and Chern classes for a (let's say) closed space $X$ is as follows: For $k = {\mathbb{R}}$ or $\mathbb{C}$, any $k$-line bundle $\xi \to X$ is the ...
1
vote
0answers
24 views

Induced measurable subbundle

Let $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ measurable and suppose that the application $$ \displaystyle{\begin{array}{rccl} h:&Z&\longrightarrow& ...
3
votes
0answers
91 views

measurable function induces a measurable bundle

Greetings I am preparing a work on bundles and I found this statement Let $V$ a topological vector space with $\dim V=n$ and $(E,\pi,M)$ a vector bundle continuous over $M$ (compact space). If ...
0
votes
1answer
19 views

generation of sub bundle

Let $M$ differentiable manifold with $\dim M=n$. If $(TM,\pi,M)$ be the fiber bundle tangent. Consider the family $E=\lbrace E_x\rbrace _{x\in M}$ such that $E_x \subset T_xM$ and $\dim E_x=k$ for ...
0
votes
1answer
52 views

The tangent bundle over a manifold is trivial iff the manifold is paralelizable

Why is the tangent bundle over a manifold trivial if and only if the manifold is parallelizable? What additional condition do we need to impose on a fiber bundle if we want it to be trivial exactly ...
0
votes
0answers
39 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: ...
2
votes
1answer
57 views

vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
0
votes
1answer
56 views

Measurable vector bundles trivial

I hope you can help: If $E$ is a measurable vector bundle over a compact metric space $(X,\mu)$ then there is a subset $Y\subset X$ such that $\mu(Y)=1$ and $\pi ^{-1}(Y)$ is isomorphic to a trivial ...
3
votes
1answer
67 views

When does the difference between a vector bundle and the associated frame bundle matter?

In the comments to this question How a principal bundle and the associated vector bundle determine each other, it was remarked that while there is a bijective correspondence between rank $n$ vector ...
-4
votes
1answer
290 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 ...
1
vote
0answers
84 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
1
vote
1answer
44 views

Maps of $G$-bundles

A vector bundle is a $GL(\mathbb{R}^m)$-bundle with fiber $\mathbb{R}^m$. A principal $G$-bundle is a $G$-bundle with fiber $G$ (where the "$G$" in "$G$-bundle" embeds into $\mathrm{Aut}\, [G]$ by ...
1
vote
1answer
135 views

An obviously false proof that all line bundles are trivial.

Let $\mathbb F$ be a field, $B$ a topological space, and consider $\operatorname{Vect}^1_{\mathbb F}(B)$, the collection of (isomorphism classes) of line bundles over $B$. There are clearly examples ...
0
votes
0answers
56 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 ...
1
vote
1answer
89 views

Recovering an Ehresmann connection from its parallel transport

How can we recover an Ehresmann connection on a general fiber bundle (as a horizontal distribution) knowing only its induced parallel transport?
4
votes
2answers
260 views

Bundle Automorphisms, Structure Groups and Gauge Groups

I am trying to get my head around the mathematical foundations of gauge theory and wanted to check that I am correct in thinking the following is true. If $E$ is a $G$-principle bundle over $M$ then ...
2
votes
1answer
222 views

Triviality/non-triviality of line/circle bundle over $S^3$

I am interested in whether I can find non-trivial bundles in the case of a fiber bundle with base space $S^3$ and fiber either $\mathbb{R}$ or $S^1$. I know that in the case of a principal bundle ...
3
votes
1answer
180 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 ...
3
votes
3answers
1k views

Understanding the trivialisation of a normal bundle

I've been looking for a definition of "trivialisation of normal bundle". I think I understand what a vector bundle, fibre bundle and a local trivialisation of either is. I also know what a tangent ...
5
votes
0answers
120 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 ...
0
votes
1answer
162 views

Is a sub-bundle of a vector bundle a vector bundle?

Could anyone please help me with this question? (1) Let (E, p, B) be a vector bundle where E is the total space, B is the base, and p is the structure map, that is, p:E->B. Now suppose E' is a ...
13
votes
7answers
594 views

An example of a triple $(E,\pi,M)$ which is not a vector bundle

What is an example of a pair of finite dimensional $C^{\infty}$ manifolds $E$ and $M$, and a smooth function $\pi:E\rightarrow M$ such that $\pi^{-1}(p)$ has a vector space structure for each $p\in M\ ...
1
vote
2answers
94 views

Does the section and the bottom manifold equivalent in the bundle?

Here the equivalent means:in the bundle ,B is bottom manifold(also called base space),S is section,we can also treat S as a bottom and B as a section in the same bundle. Is it true in the vector ...