Double vector bundle

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

A double vector bundle consists of ${\displaystyle (E,E^{H},E^{V},B)}$, where

...
Wikipedia