In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent T E {\displaystyle TE} of a vector bundle E {\displaystyle E} and the double tangent bundle T 2 M {\displaystyle T^{2}M}.

Definition and first consequences

A double vector bundle consists of ( E , E H , E V , B ) {\displaystyle (E,E^{H},E^{V},B)}, where

  1. the side bundles E H {\displaystyle E^{H}} and E V {\displaystyle E^{V}} are vector bundles over the base B {\displaystyle B},
  2. E {\displaystyle E} is a vector bundle on both side bundles E H {\displaystyle E^{H}} and E V {\displaystyle 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 ) {\displaystyle (f_{E},f_{H},f_{V},f_{B})} consists of maps f E : E ↦ E ′ {\displaystyle f_{E}:E\mapsto E'}, f H : E H ↦ E H ′ {\displaystyle f_{H}:E^{H}\mapsto E^{H}{}'}, f V : E V ↦ E V ′ {\displaystyle f_{V}:E^{V}\mapsto E^{V}{}'} and f B : B ↦ B ′ {\displaystyle f_{B}:B\mapsto B'} such that ( f E , f V ) {\displaystyle (f_{E},f_{V})} is a bundle morphism from ( E , E V ) {\displaystyle (E,E^{V})} to ( E ′ , E V ′ ) {\displaystyle (E',E^{V}{}')}, ( f E , f H ) {\displaystyle (f_{E},f_{H})} is a bundle morphism from ( E , E H ) {\displaystyle (E,E^{H})} to ( E ′ , E H ′ ) {\displaystyle (E',E^{H}{}')}, ( f V , f B ) {\displaystyle (f_{V},f_{B})} is a bundle morphism from ( E V , B ) {\displaystyle (E^{V},B)} to ( E V ′ , B ′ ) {\displaystyle (E^{V}{}',B')} and ( f H , f B ) {\displaystyle (f_{H},f_{B})} is a bundle morphism from ( E H , B ) {\displaystyle (E^{H},B)} to ( E H ′ , B ′ ) {\displaystyle (E^{H}{}',B')}.

The 'flip of the double vector bundle ( E , E H , E V , B ) {\displaystyle (E,E^{H},E^{V},B)} is the double vector bundle ( E , E V , E H , B ) {\displaystyle (E,E^{V},E^{H},B)}.

Examples

If ( E , M ) {\displaystyle (E,M)} is a vector bundle over a differentiable manifold M {\displaystyle M} then ( T E , E , T M , M ) {\displaystyle (TE,E,TM,M)} is a double vector bundle when considering its secondary vector bundle structure.

If M {\displaystyle M} is a differentiable manifold, then its double tangent bundle ( T T M , T M , T M , M ) {\displaystyle (TTM,TM,TM,M)} is a double vector bundle.

Mackenzie, K. (1992), "Double Lie algebroids and second-order geometry, I", Advances in Mathematics, 94 (2): 180–239, doi: