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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
In other words, a characteristic class associates to each principal G-bundle in () an element c(P) in H*(X) such that, if f : Y → X is a continuous map, then c(f*P) = f*c(P). On the left is the class of the pullback of P to Y ; on the right is the image of the class of P under the induced map in cohomology.
A section of a tangent vector bundle is a vector field. A vector bundle E {\displaystyle E} over a base M {\displaystyle M} with section s {\displaystyle s} . In the mathematical field of topology , a section (or cross section ) [ 1 ] of a fiber bundle E {\displaystyle E} is a continuous right inverse of the projection function π ...
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.
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.
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.