  In this paper we propose a novel second-order accurate well balanced scheme
for shallow water equations in general covariant coordinates over manifolds. In
our approach, once the gravitational field is defined for the specific case,
one equipotential surface is detected and parametrized by a frame of general
covariant coordinates. This surface is the manifold whose covariant
parametrization induces a metric tensor. The model is then re-written in a
hyperbolic form with a tuple of conserved variables composed both of the
evolving physical quantities and the metric coefficients. This formulation
allows the numerical scheme to automatically compute the curvature of the
manifold as long as the physical variables are evolved.
