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

  • Context: Graduate 
  • Thread starter Thread starter robforsub
  • Start date Start date
  • Tags Tags
    Dual Vector
Click For Summary
SUMMARY

The dual vector bundle E* is defined as Hom(E, MXR), establishing a direct isomorphism between E* and Hom(E, MXR). The discussion highlights the natural isomorphism Hom(E, E') = E* ⊕ E, which can be leveraged to demonstrate that E* is isomorphic to E* ⊕ MXR. The fibers of the dual vector bundle E* are the vector space duals of the fibers of E, confirming that the definition of E* aligns with standard mathematical conventions.

PREREQUISITES
  • Understanding of vector bundles and their properties
  • Familiarity with the concept of dual spaces in linear algebra
  • Knowledge of Hom functors in category theory
  • Basic comprehension of fiber bundles and their structure
NEXT STEPS
  • Study the properties of vector bundles in differential geometry
  • Explore the concept of fiber-wise duals in vector spaces
  • Learn about the applications of Hom functors in algebraic topology
  • Investigate the implications of isomorphisms in vector bundle theory
USEFUL FOR

Mathematicians, particularly those specializing in differential geometry and algebraic topology, as well as graduate students studying vector bundles and their duals.

robforsub
Messages
16
Reaction score
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)?
 
Physics news on Phys.org
So how do you define E*?
 
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)?
 
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).
 
I'm failing to see how it's not tautological.
 

Similar threads

Undergrad Dot product between basis vectors and dual basis vectors
  • · Replies 17 ·
Replies
17
Views
2K
Graduate Problem in Understanding notation of distributional section
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Identification tangent bundle over affine space with product bundle
  • · Replies 13 ·
Replies
13
Views
3K
Graduate Two questions for vector bundles
  • · Replies 18 ·
Replies
18
Views
2K
Undergrad "Adding" a Vector Space and its Dual
  • · Replies 16 ·
Replies
16
Views
3K
High School A covariant vs contravariant vector?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Understanding the concepts of isometric basis and musical isomorphism
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Inner product vs dot/scalar product
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Transformation of coordinate basis
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Galilean spacetime as a fiber bundle
  • · Replies 16 ·
Replies
16
Views
5K