Double vector bundle explained

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

TE of a vector bundle E and the double tangent bundle T ^{2M} .

Definition and first consequences

A double vector bundle consists of

(E, E ^{H,} E ^{V,} B) , where

1. the side bundles

E ^{H} and E ^{V} are vector bundles over the base B ,E is a vector bundle on both side bundles E ^{H} and E ^{V} ,

1. 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

(f _{E,} f _{H,} f _{V,} f _{B)} consists of maps f _{E} : E \mapsto E' , f _{H} : E ^{H} \mapsto E ^{H{}'} , f _{V} : E ^{V} \mapsto E ^{V{}'} and f _{B} : B \mapsto B' such that (f _{E,} f _{V)} is a bundle morphism from (E, E ^{V)} to (E', E ^{V{}')} , (f _{E,} f _{H)} is a bundle morphism from (E, E ^{H)} to (E', E ^{H{}')} , (f _{V,} f _{B)} is a bundle morphism from (E ^{V,} B) to (E ^{V{}',} B') and (f _{H,} f _{B)} is a bundle morphism from (E ^{H,} B) to (E ^{H{}',} B') .

The flip of the double vector bundle

(E, E ^{H,} E ^{V,} B) is the double vector bundle (E, E ^{V,} E ^{H,} B) .

Examples

If

(E, M) is a vector bundle over a differentiable manifold M then (TE, E, TM, M) is a double vector bundle when considering its secondary vector bundle structure.

If

M is a differentiable manifold, then its double tangent bundle (TTM, TM, TM, M) is a double vector bundle