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.
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. It becomes clear actually defines addition on the fibers of as . The triple becomes a smooth vector bundle with these vector space operations on its fibres.
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.
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 .