Secondary vector bundle structure


In mathematics, particularly differential topology, the secondary vector bundle structure
refers to the natural vector bundle structure on the total space TE of the tangent bundle of a smooth vector bundle, induced by the push-forward of the original projection map.
This gives rise to a double vector bundle structure.
In the special case, where is the double tangent bundle, the secondary vector bundle is isomorphic to the tangent bundle
of through the canonical flip.

Construction of the secondary vector bundle structure

Let be a smooth vector bundle of rank. Then the preimage of any tangent vector in in the push-forward of the canonical projection is a smooth submanifold of dimension, and it becomes a vector space with the push-forwards
of the original addition and scalar multiplication
as its vector space operations. The triple becomes a smooth vector bundle with these vector space operations on its fibres.

Proof

Let be a local coordinate system on the base manifold with and let
be a coordinate system on adapted to it. Then
so the fiber of the secondary vector bundle structure at in is of the form
Now it turns out that
gives a local trivialization for, and the push-forwards of the original vector space operations read in the adapted coordinates as
and
so each fibre is a vector space and the triple is a smooth vector bundle.

Linearity of connections on vector bundles

The general Ehresmann connection on a vector bundle can be characterized in terms of the connector map
where is the vertical lift, and is the vertical projection. The mapping
induced by an Ehresmann connection is a covariant derivative on in the sense that
if and only if the connector map is linear with respect to the secondary vector bundle structure on. Then the connection is called linear. Note that the connector map is automatically linear with respect to the tangent bundle structure.