  First order Hamiltonian operators of differential-geometric type were
introduced by Dubrovin and Novikov in 1983, and thoroughly investigated by
Mokhov. In 2D, they are generated by a pair of compatible flat metrics $g$ and
$\tilde g$ which satisfy a set of additional constraints coming from the
skew-symmetry condition and the Jacobi identity. We demonstrate that these
constraints are equivalent to the requirement that $\tilde g$ is a linear
Killing tensor of $g$ with zero Nijenhuis torsion. This allowed us to obtain a
complete classification of $n$-component operators with $n\leq 4$ (for $n=1, 2$
this was done before). For 2D operators the Darboux theorem does not hold: the
operator may not be reducible to constant coefficient form. All interesting
(non-constant) examples correspond to the case when the flat pencil $g, \tilde
g$ is not semisimple, that is, the affinor $\tilde g g^{-1}$ has non-trivial
Jordan block structure. In the case of a direct sum of Jordan blocks with
distinct eigenvalues we obtain a complete classification of Hamiltonian
operators for any number of components $n$, revealing a remarkable
correspondence with the class of trivial Frobenius manifolds modelled on
$H^*(CP^{n-1})$.
