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Vector bundle morphisms are a special case of the notion of a bundle map between fiber bundles, and are sometimes called (vector) bundle homomorphisms. A bundle homomorphism from E 1 to E 2 with an inverse which is also a bundle homomorphism (from E 2 to E 1) is called a (vector) bundle isomorphism, and then E 1 and E 2 are said to be ...
If E is a complex vector bundle, then the conjugate bundle ¯ of E is obtained by having complex numbers acting through the complex conjugates of the numbers. Thus, the identity map of the underlying real vector bundles: ¯ = is conjugate-linear, and E and its conjugate E are isomorphic as real vector bundles.
From any vector bundle, one can construct the frame bundle of bases, which is a principal bundle (see below). Another special class of fiber bundles, called principal bundles , are bundles on whose fibers a free and transitive action by a group G {\displaystyle G} is given, so that each fiber is a principal homogeneous space .
In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration .
Pages in category "Vector bundles" The following 55 pages are in this category, out of 55 total. This list may not reflect recent changes. ...
Given a vector bundle of rank , and any representation : (,) into a linear group (), there is an induced connection on the associated vector bundle =. This theory is most succinctly captured by passing to the principal bundle connection on the frame bundle of E {\displaystyle E} and using the theory of principal bundles.
In the case of projective space, where the tautological bundle is a line bundle, the associated invertible sheaf of sections is (), the tensor inverse (ie the dual vector bundle) of the hyperplane bundle or Serre twist sheaf (); in other words the hyperplane bundle is the generator of the Picard group having positive degree (as a divisor) and ...
Given a morphism : of vector bundles over the same space, there is a morphism : between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map : (). Accordingly, the dual bundle operation defines a contravariant functor from the category of vector bundles and their morphisms to itself.