Double vector bundle

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 ,
2. E is a vector bundle on both side bundles E H and E V ,
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 (f_E, f_H, f_V, f_B) consists of maps f E : E ↦ E ′ , f H : E H ↦ E H ′ , f V : E V ↦ E V ′ and f B : B ↦ 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 ( T E , E , T M , M ) is a double vector bundle when considering its secondary vector bundle structure.

If M is a differentiable manifold, then its double tangent bundle ( T T M , T M , T M , M ) is a double vector bundle.