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
 * 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 $$f_E : E \mapsto E'$$, $$f_H : E^H \mapsto E^H{}'$$,  $$f_V : E^V \mapsto E^V{}'$$ 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.