Double vector bundle vs vector bundle over supermanifolds

Question

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.

This post imported from StackExchange MathOverflow at 2014-08-10 19:52 (UCT), posted by SE-user Ma Ming

Comments

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?

This post imported from StackExchange MathOverflow at 2014-08-10 19:52 (UCT), posted by SE-user მამუკა ჯიბლაძე