Double vector bundle

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

Definition and first consequences

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

1. the side bundles [math]\displaystyle{ E^H }[/math] and [math]\displaystyle{ E^V }[/math] are vector bundles over the base [math]\displaystyle{ B }[/math],
2. [math]\displaystyle{ E }[/math] is a vector bundle on both side bundles [math]\displaystyle{ E^H }[/math] and [math]\displaystyle{ E^V }[/math],
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 [math]\displaystyle{ (f_E, f_H, f_V, f_B) }[/math] consists of maps [math]\displaystyle{ f_E : E \mapsto E' }[/math], [math]\displaystyle{ f_H : E^H \mapsto E^H{}' }[/math], [math]\displaystyle{ f_V : E^V \mapsto E^V{}' }[/math] and [math]\displaystyle{ f_B : B \mapsto B' }[/math] such that [math]\displaystyle{ (f_E, f_V) }[/math] is a bundle morphism from [math]\displaystyle{ (E, E^V) }[/math] to [math]\displaystyle{ (E', E^V{}') }[/math], [math]\displaystyle{ (f_E, f_H) }[/math] is a bundle morphism from [math]\displaystyle{ (E, E^H) }[/math] to [math]\displaystyle{ (E', E^H{}') }[/math], [math]\displaystyle{ (f_V, f_B) }[/math] is a bundle morphism from [math]\displaystyle{ (E^V, B) }[/math] to [math]\displaystyle{ (E^V{}', B') }[/math] and [math]\displaystyle{ (f_H, f_B) }[/math] is a bundle morphism from [math]\displaystyle{ (E^H, B) }[/math] to [math]\displaystyle{ (E^H{}', B') }[/math].

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

Examples

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

If [math]\displaystyle{ M }[/math] is a differentiable manifold, then its double tangent bundle [math]\displaystyle{ (TTM, TM, TM , M) }[/math] 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