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

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 $f_{E}:E\mapsto E'$, $f_{H}:E^{H}\mapsto E^{H}{}'$, $f_{V}:E^{V}\mapsto E^{V}{}'$ 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.

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