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 $T^2M$.

Definition and first consequences

A double vector bundle consists of $(E, E^H, E^V, B)$, where
# the ''side bundles'' $E^H$ and $E^V$ are vector bundles over the base $B$,
# $E$ is a vector bundle on both side bundles $E^H$ and $E^V$,
# 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

{{Citation
, last = Mackenzie
, first = K.
, title = Double Lie algebroids and second-order geometry, I
, journal = Advances in Mathematics
, volume = 94
, number = 2
, date = 1992
, pages = 180–239
, doi=10.1016/0001-8708(92)90036-k
, doi-access = free

Differential geometry
Topology
Differential topology