0
votes
0answers
16 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 ...
4
votes
1answer
64 views

Why doesn't a metric give an isomorphism $TX \cong T^*X$?

Any smooth manifold $X$ admits a Riemannian metric $g$, and we have a map $$ TX \to T^*X, \qquad (x, v) \mapsto (x, g(v,-)) $$ which is smooth if $g$ is. Why isn't this an isomorphism of vector ...
2
votes
0answers
20 views

Direct sums, tensor products etc. of $G$-vector bundles are again $G$-spaces

Given two $G$-vector bundles $E$ and $F$ over a $G$-space $X$ ($G$ some finite group), I am interested in the vector bundles $E \oplus F$, $E \otimes F$, $\operatorname{Hom}(E,F)$ etc. I am familiar ...
1
vote
1answer
77 views

Topology of $GL_n(K)$

I need to show any of the following results: Consider $K=\mathbb{R}$ or $\mathbb{C}$, then, 1) The compact-open topology and the usual topology of $GL_n(K)$ are the same. 2) Taking inverses and ...
1
vote
1answer
49 views

Some troubles about topology and definition of a Vector Bundle

Disclaimer: It's heavily related to my old question : Visualizing the Topology of a Vector Bundle but I wanted to open a new question because the former had already got an answer and this time my ...
5
votes
1answer
131 views

Visualizing topology of a Vector Bundle

I've started reading Milnor, Stasheff - Characteristic Classes and at page $18$ they proved that $\mathbb{R}^n$-bundle $\xi$ is trivial if and only if $\xi$ admits $n$ cross sections $s_1, \dots , ...
1
vote
1answer
33 views

The set of all sections of a vector bundle

At http://en.wikipedia.org/wiki/Vector_bundle we have: "Given a vector bundle $\pi : E \rightarrow X$ and an open subset $U$ of $X$, we can consider sections of $\pi$ on $U$, i.e. continuous ...
0
votes
0answers
28 views

The homotopy between two monomorphisms

$X$ is a compact Hausdorff space and $E,F$ are two complex vector bundles on $X$. If $f$ and $g$ are two homotopic monomorphisms from $E$ to $F$, then can we find a homotopy $f_t$ such that $f_0=f, ...
1
vote
1answer
56 views

Doubt on how to prove proposition about bundles

I've started studying bundles and fiber bundles and to get some practice I've tried to prove the following proposition: "Every vector bundle $(E,B,\pi,F,G)$ is associated to a given principal bundle ...
4
votes
1answer
322 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 ...
3
votes
1answer
157 views

Pontryagin classes of a product manifold

I'm imagining there's a way to relate the pontryagin classes of $T(M\times N)$ to the pontryagin classes of $M$ and those of $N$, but I haven't been able to find a helpful reference. Could someone ...
0
votes
1answer
158 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 ...
10
votes
1answer
141 views

Is there a characterization of injective $C(X)$-modules analogous to Serre-Swan?

The Serre-Swan theorem in topology says that if $X$ is compact Hausdorff and $C(X)$ the ring of continuous functions on $X$, then the category of finitely generated projective $C(X)$-modules is ...
5
votes
1answer
200 views

Pasting Together Fibers of a Vector Bundle

Everyone: Please forgive that I do not yet know LaTex, bro, and my English ( I am from UCV in Venezuela). I think I understand concept of bundles almost well, and that, once a vector bundle with a ...