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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 ...
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 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 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 ...
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 ...
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 ...
In mathematics, an adjoint bundle [1] is a vector bundle naturally associated with any smooth principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles have important applications in the theory of connections as well as in gauge theory.
This principal bundle is called the holonomy bundle (through p) of the connection. The connection ω restricts to a connection on H ( p ), since its parallel transport maps preserve H ( p ). Thus H ( p ) is a reduced bundle for the connection.