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 T2M.

## Contents

* 1 Definition and first consequences
* 2 Double vector bundle morphism
* 3 Examples
* 4 References

## Definition and first consequences

A double vector bundle consists of (E,EH,EV,B), where

1. the side bundles EH and EV are vector bundles over the base B,
2. E is a vector bundle on both side bundles EH and EV,
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 (fE,fH,fV,fB) consists of maps fE:E↦E′, fH:EH↦EH′, fV:EV↦EV′ and fB:B↦B′ such that (fE,fV) is a bundle morphism from (E,EV) to (E′,EV′), (fE,fH) is a bundle morphism from (E,EH) to (E′,EH′), (fV,fB) is a bundle morphism from (EV,B) to (EV′,B′) and (fH,fB) is a bundle morphism from (EH,B) to (EH′,B′).

The 'flip of the double vector bundle (E,EH,EV,B) is the double vector bundle (E,EV,EH,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

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