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 Lagrang 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 $\nabla$?