About: Secondary vector bundle structure

In mathematics, particularly differential topology, the secondary vector bundle structurerefers 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 TE = TTM 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.