Quasi-riemannian structures on supermanifolds and characteristic classes

The notion of a quasi-Riemannian metric being an alternative to generalization of the Riemann metrics to supermanifolds is introduced. Unlike standard supermetrics, the quasi-Riemannian metrics exist on arbitrary supermanifolds, though they are not supersymmetric under the permutation of indices. The application of the quasi-Riemannian structures to the theory of characteristic classes of supermanifolds is considered.