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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 ...
In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: E → B with E and B sets.
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 ...
In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres. Definition ...
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.
A Koszul connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead to convenient formulations of geometric invariants , such as the curvature (see also curvature tensor and curvature form ), and torsion tensor .