[1] A note on notation: in this article, we denote projection maps by their domains, e.g., πTTM : TTM → TM.
There is also a stronger result of this kind [4] which states that if N is a 2n-dimensional manifold and if there exists a (1,1)-tensor field J on N that satisfies then N is diffeomorphic to an open set of the total space of a tangent bundle of some n-dimensional manifold M, and J corresponds to the tangent structure of TM in this diffeomorphism.
In any associated coordinate system on TM the canonical vector field and the canonical endomorphism have the coordinate representations A Semispray structure on a smooth manifold M is by definition a smooth vector field H on TM \0 such that JH=V.
The canonical flip makes it possible to define nonlinear covariant derivatives on smooth manifolds as follows.
Looking at the local representations one can confirm that the Ehresmann connections on (TM/0,πTM/0,M) and nonlinear covariant derivatives on M are in one-to-one correspondence.