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robforsub
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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)?
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.
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.
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.
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.
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.