Dual vector bundle E* is isomorphic to Hom(E, MXR)

In summary, the conversation discusses the definition and isomorphism of dual vector bundles, specifically E* and Hom(E, MXR). The question arises about the possibility of using the isomorphism Hom(E,E')=E*(direct sum)E to show that E* is isomorphic to E*(direct sum) MXR and thus Hom(E, MXR). It is mentioned that for the speaker, the dual of a vector bundle E is obtained by taking fiber-wise duals, and the dual of a k-vector space Ep is Ep*=Hom_k(E,k). The conversation ends with confusion about the tautological nature of the discussion.
  • #1
robforsub
16
0
As E* is defined in some book as Hom(E, MXR). What could be the isomorphism between dual vector bundle E* and Hom(E, MXR)?
 
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  • #2
So how do you define E*?
 
  • #3
That's the part that is confusing to me. So I have checked on wikipedia, and it defines E*=Hom(E,MXR). However, there is a natural isomorphism on bundle that is Hom(E,E')=E*(direct sum)E, therefore I am wondering if I can use this isomorphism to get the result that E* is isomorphic to E*(direct sum) MXR and thus isomorphic to Hom(E, MXR)?
 
  • #4
Still, how does your question even make sense if you have no definition of E*? For me, the dual of a vector bundle E is obtained from E by taking fiber-wise duals, i.e. the fibers of E* are the vector space duals of the fibers of E. Of course, the dual of a k-vector space Ep is Ep*=Hom_k(E,k).
 
  • #5
I'm failing to see how it's not tautological.
 

1. What is a dual vector bundle?

A dual vector bundle is a mathematical object that consists of two vector bundles, one over the base space and one over the dual space. It is used to describe the relationship between a vector bundle and its dual bundle.

2. What does it mean for E* to be isomorphic to Hom(E, MXR)?

When E* (the dual vector bundle) is isomorphic to Hom(E, MXR) (the bundle of linear maps from E to the constant bundle MXR), it means that there is a one-to-one correspondence between the two bundles. This allows for the dual vector bundle to be described in terms of Hom(E, MXR) and vice versa.

3. What is the significance of this isomorphism?

The isomorphism between E* and Hom(E, MXR) allows for a deeper understanding of the relationship between a vector bundle and its dual bundle. It also allows for more efficient calculations and proofs in mathematical and scientific applications.

4. How is this isomorphism used in physics?

In physics, this isomorphism is used to describe the relationship between vector fields and differential forms. It is also used in theories such as gauge theory, where the gauge fields can be represented as sections of a vector bundle and their dual fields can be represented as sections of the dual vector bundle.

5. Can this isomorphism be extended to other mathematical objects?

Yes, the concept of duality and isomorphism between a vector bundle and its dual bundle can be extended to other mathematical objects, such as sheaves and cohomology groups. However, the specific isomorphism between E* and Hom(E, MXR) is unique to vector bundles.

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