Secondary vector bundle structure

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 $p * : TE \to TM$ of the original projection map $p : E \to M$ . This gives rise to a double vector bundle structure $(TE, E, TM, M)$ .

In the special case $(E, p, M) = (TM, π TM, M)$ , where $TE = TTM$ is the double tangent bundle, the secondary vector bundle $(TTM, (π TM) *, TM)$ is isomorphic to the tangent bundle $(TTM, π TTM, TM)$ of $TM$ through the canonical flip.

Let $(E, p, M)$ be a smooth vector bundle of rank $N$ . Then the preimage $(p *) -1 (X) \subset TE$ of any tangent vector $X$ in $TM$ in the push-forward $p * : TE \to TM$ of the canonical projection $p : E \to M$ is a smooth submanifold of dimension $2 N$ , 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.

Let $(U, φ)$ be a local coordinate system on the base manifold $M$ with $φ (x) = (x 1, ..., x n)$ and let

...
Wikipedia