# Tagged Questions

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### 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 ...
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### 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 ...
0answers
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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335 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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...