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) . In the special case (E, p, M) = (TM, πTM, M) , where is the double tangent bundle, the secondary vector bundle (TTM, (πTM)∗, TM) is isomorphic to the tangent bundle (TTM, πTTM, TM) of TM through the canonical flip.

Construction of the secondary vector bundle structure

Let (E, p, M) be a smooth vector bundle of rank N. Then the preimage (p∗)−1(X) ⊂ TE of any tangent vector X in TM in the push-forward p∗ : TE → TM of the canonical projection p : E → M is a smooth submanifold of dimension 2N , and it becomes a vector space with the push-forwards of the original addition and scalar multiplication as its vector space operations. The triple (TE, p∗, TM) becomes a smooth vector bundle with these vector space operations on its fibres.

Proof

Let (U, φ) be a local coordinate system on the base manifold M with and let be a coordinate system on adapted to it. 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.

Linearity of connections on vector bundles

The general Ehresmann connection 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 . Then the connection is called linear. Note that the connector map is automatically linear with respect to the tangent bundle structure (TE, πTE, E) .