A criterion for foliations with transverse linear connection to be pseudo-Riemannian

We obtain necessary and sufficient conditions for a foliation of codimension $q$ on $n$-dimensional manifold with transverse linear connection to admit a transverse invariant pseudo-Riemannian metric of a given signature which is parallel with the respect to the indicated connection. In particular we obtain a criterion for a foliation with transverse linear connection to be Riemannian foliation.