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In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space (for example could be a topological space, a manifold, or an algebraic variety): to every point of the space we associate (or "attach") a vector space () in such a way that these vector spaces fit ...
One example of a principal bundle is the frame bundle. If for each two points b 1 and b 2 in the base, the corresponding fibers p −1 (b 1) and p −1 (b 2) are vector spaces of the same dimension, then the bundle is a vector bundle if the appropriate conditions of local triviality are satisfied. The tangent bundle is an example of a vector ...
A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle B 1 := (M × N, pr 1) with bundle projection pr 1 : M × N → M : (x, y) → x. Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in M × N. Then the image of this point under pr 1 is m
Important examples of vector bundles include the tangent bundle and cotangent bundle of a smooth manifold. 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 ...
Consider the sphere as the union of the upper and lower hemispheres + and along their intersection, the equator, an .. Given trivialized fiber bundles with fiber and structure group over the two hemispheres, then given a map : (called the clutching map), glue the two trivial bundles together via f.
Examples for vector bundles include: the introduction of a metric resulting in reduction of the structure group from a general linear group to an orthogonal group (); and the existence of complex structure on a real bundle resulting in reduction of the structure group from real general linear group (,) to complex general linear group (,).
For example, when is a vector bundle a section of is an element of the vector space lying over each point . In particular, a vector field on a smooth manifold M {\displaystyle M} is a choice of tangent vector at each point of M {\displaystyle M} : this is a section of the tangent bundle of M {\displaystyle M} .
As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space X to another topological space Y is associated the pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image ...