Double vector bundle

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

Definition and first consequences

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

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

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

Examples

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

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

References

Mackenzie, K. (1992), "Double Lie algebroids and second-order geometry", Adv. Math., 94 (2): 180–239, doi:10.1016/0001-8708(92)90036-k