  Based on the theory of Poisson vertex algebras we calculate skew-symmetry
conditions and Jacobi identities for a class of third-order nonlocal operators
of differential-geometric type. Hamiltonian operators within this class are
defined by a Monge metric and a skew-symmetric two-form satisfying a number of
differential-geometric constraints. Complete classification results in the
2-component and 3-component cases are obtained.
