# Double vector bundle vs vector bundle over supermanifolds

Double vector bundle is roughly a vector bundle over (horizontal) vector bundle.

Vector bundle is a supermanifold in a nature way (non-nature on the other way around).

My question: is a double vector bundle a vector bundle over the supermanifold corresponding to the horizontal vector bundle in a nature way? I can see this is true for double tangent bundle.

• Can you please tell in more detail what are these supermanifold structures on (presumably total space of) a vector bundle, and in what sense are they related to nature/non-nature? – მამუკა ჯიბლაძე Mar 19 '14 at 5:37