Assume that $M$ is a $n$ dimensional Riemannian manifold. So $TM$ has a natural structure of a symplectic manifold. A Lagrangian connection $D$ on $M$ is a $n$ dimensional distribution for $TM$ such that for all $z\in TM$, $D_z$ is a lagrangian subspace of $T_z TM$ which is transverse to the vertical foltion of $TM$.

Does every manifold admit a Lagrangian connection? Is the $LC$ connection necessarily a Lagranģian connection?

Let $\nabla$ be the corresponding derivation associated with a Lagrangian connection.

What formula is satisfied by $\nabla$?