When a coordinate systemξ:p7→ξp= (ξ1p, . . . ,ξnp)∈Rnis given inM, we pose one condition that, for two nearby pointsξpandξq=ξp+∆ξ,Dis expanded asD(p‖q) =12Dgij(p)∆ξi∆ξj+O(‖∆ξ‖3)(2)and(Dgij(p))ijis a positive definite matrix.

A divergence functionD(p‖q)is a differentiable real-valued function of two pointspandqin amanifoldM. It satisfies the non-negativity conditionD(p‖q)≥0(1)with equality if and only ifp=q.