Relationship between $\Gamma (E)\otimes \Gamma (B) $ and $\Gamma(E\otimes B) $

Question

I have a question that I've been thinking about for a while. So If $(E,\pi_E)$ and $(B,\pi_B)$ are some vector bundles over some mainfold $M$. What is the exact relationship between $$\Gamma (E)\otimes \Gamma (B) $$ and $$\Gamma(E\otimes B) $$

Are they the same spaces, or if not, are they at least isomorphic to one another?

possible duplicate of Global sections of a tensor product of vector bundles on a smooth manifold — anomaly, Feb 10 '15 at 23:02
Try using the fact that bundles over a (reasonable) manifold are stably trivial. — anomaly, Feb 10 '15 at 23:02
Here is the answer talking about relations between various sections of vector bundles
http://math.stackexchange.com/questions/1323787/what-is-the-relation-between-c-infty-linear-and-tensorial/1323833#1323833 — Fallen Apart, Sep 07 '15 at 18:13

score 3 · Accepted Answer · answered Feb 10 '15 at 23:39

3

The natural map goes this way $\Gamma(E) \otimes \Gamma(F) \to \Gamma(E\otimes F)$.

Injectiveness is not a problem.

Surjectiveness is. Think of compact connected complex manifold, $E$ a bundle with a $0$ space of global sections, while $E\otimes F$ has lots of global sections.

However in the smooth or topological category things are OK because it is true for trivial bundles and it is true for a direct sum if and only if it is true for summands. Moreover, every bundle is a direct summand of a trivial one, like @anomaly: indicated. In fact, bundles are equivalent to finitely generated projective module over the ring of functions on the manifold (Swan's theorem) and things work fine for tensor products.

answered Feb 10 '15 at 23:39

orangeskid

53,909

So that's why in Jost's book, he's able to say that $$\Omega^2(E) \otimes (\Omega^0(E))^* = \Omega^2(End(E)) $$ – MonkSphere Feb 11 '15 at 20:53
@MonkSphere: Yes, I think so. Locally it is correct. But one cannot globalize. I imagine you can stick this into an exact sequence - the specialists can tell. The smooth or topological category is very supple. Or, work over an affine ( Stein) base space, then things are OK again. – orangeskid Feb 11 '15 at 21:03
@orangeskid The isomorphism you describe for trivial bundles is not canonical thought, correct? It involves choosing frames for both bundles? – user2520938 Apr 29 '16 at 11:45

Relationship between $\Gamma (E)\otimes \Gamma (B) $ and $\Gamma(E\otimes B) $

1 Answers1

Linked