A vector bundle is a bundle with an additional vector space structure on each fibre. The concept arose from the study of tangent vector fields to smooth geometric objects, e.g., spheres, projective spaces, and, more generally, manifolds. The vector bundle structure is so rich that the set of isomorphism classes of k-dimensional vector bundles over a paracompact space B is in a natural bijective correspondence with the set of homotopy classes of mappings of B into the Grassmann manifold of k-dimensional subspaces in infinite-dimensional space.
KeywordsVector Bundle Open Covering Homotopy Class Finite Type Constant Rank
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