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Double vector bundle

In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent of a vector bundle and the double tangent bundle .

Definition and first consequences

A double vector bundle consists of , where

  1. the side bundles and are vector bundles over the base ,
  2. is a vector bundle on both side bundles and ,
  3. the projection, the addition, the scalar multiplication and the zero map on E for both vector bundle structures are morphisms.

Double vector bundle morphism

A double vector bundle morphism consists of maps , , and such that is a bundle morphism from to , is a bundle morphism from to , is a bundle morphism from to and is a bundle morphism from to .

The flip of the double vector bundle is the double vector bundle .

Examples

If is a vector bundle over a differentiable manifold then is a double vector bundle when considering its secondary vector bundle structure.

If is a differentiable manifold, then its double tangent bundle is a double vector bundle.

References