Capacities as conformal quasi-invariants on pseudo-riemannian manifolds

A class of capacities is introduced on pseudo-riemannian manifolds. They arise as a natural counterpart of the well-known plane quasiconformal capacities and their higher dimensional analogues which have been studied extensively in the recent years by F.W. Gehring, R. Kühnau and others. The capacities in question are shown to be either conformal invariants or conformal quasi-invariants, and, in the latter case, exact bounds are established. We thus arrive at the notion of quasiconformal mappings of pseudo-riemannian manifolds, which correspond to the inhomogeneous media. These mappings are studied briefly and the physical interpretation of some of the capacities in question is also given.