Double vector bundle

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 $T^{2}M$.

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 $f_E:E\mapsto E^{\prime }$, $f_H:E^H\mapsto E^H{\prime }^{}$, $f_V:E^V\mapsto E^V{\prime }^{}$ and $f_B:B\mapsto B^{\prime }$ such that $(f_E,f_V)$ is a bundle morphism from $(E,E^V)$ to $(E^{\prime },E^V{\prime }^{})$, $(f_E,f_H)$ is a bundle morphism from $(E,E^H)$ to $(E^{\prime },E^H{\prime }^{})$, $(f_V,f_B)$ is a bundle morphism from $(E^V,B)$ to $(E^V{\prime }^{},B^{\prime })$ and $(f_H,f_B)$ is a bundle morphism from $(E^H,B)$ to $(E^H{\prime }^{},B^{\prime })$.

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, I", Advances in Mathematics 94 (2): 180–239, doi:10.1016/0001-8708(92)90036-k