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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 mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. [1] This is equivalent to: A connection for which the covariant derivatives of the metric on E ...
Let : be a fibre bundle with fibre .Let be a collection of pairs (,) such that : is a local trivialization of over .Moreover, we demand that the union of all the sets is (i.e. the collection is an atlas of trivializations =).
On the other hand there was now a framework which produced families of classes, whenever there was a vector bundle involved. The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G.
be a rank n real vector bundle over the paracompact space B. Then for each point b in B, the fiber is an n-dimensional real vector space. We can form an n-sphere bundle by taking the one-point compactification of each fiber and gluing them together to get the total space.
The dual bundle of a vector bundle : is the vector bundle : whose fibers are the dual spaces to the fibers of . Equivalently, E ∗ {\displaystyle E^{*}} can be defined as the Hom bundle H o m ( E , R × X ) , {\displaystyle \mathrm {Hom} (E,\mathbb {R} \times X),} that is, the vector bundle of morphisms from E {\displaystyle E} to the trivial ...