In mathematics, particularly differential topology, the secondary vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle (E, p, M), induced by the push-forward p∗ : TE → TM of the original projection map p : E → M. This gives rise to a double vector bundle structure (TE,E,TM,M).
Then so the fiber of the secondary vector bundle structure at X in TxM is of the form Now it turns out that gives a local trivialization χ : TW → TU × R2N for (TE, p∗, TM), and the push-forwards of the original vector space operations read in the adapted coordinates as and so each fibre (p∗)−1(X) ⊂ TE is a vector space and the triple (TE, p∗, TM) is a smooth vector bundle.
The general Ehresmann connection TE = HE ⊕ VE on a vector bundle (E, p, M) can be characterized in terms of the connector map where vlv : E → VvE is the vertical lift, and vprv : TvE → VvE is the vertical projection.
The mapping induced by an Ehresmann connection is a covariant derivative on Γ(E) in the sense that if and only if the connector map is linear with respect to the secondary vector bundle structure (TE, p∗, TM) on TE.
Note that the connector map is automatically linear with respect to the tangent bundle structure (TE, πTE, E).