Double vector bundle

In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent

{\displaystyle TE}

of a vector bundle

{\displaystyle E}

and the double tangent bundle

2

{\displaystyle T^{2}M}

.

A double vector bundle consists of

(

,

,

,

)

, where A double vector bundle morphism

(

consists of maps

is a bundle morphism from

is a bundle morphism from

is a bundle morphism from

is a bundle morphism from

The 'flip of the double vector bundle

is the double vector bundle

is a vector bundle over a differentiable manifold

{\displaystyle (TE,E,TM,M)}

is a double vector bundle when considering its secondary vector bundle structure.

is a differentiable manifold, then its double tangent bundle

{\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:10.1016/0001-8708(92)90036-k