We derive new models for gravity driven shallow water flows
in several space dimensions over a general topography.
A first model is valid for small slope variation, i.e.
small curvature, and a second model is valid for arbitrary
topography. In both cases no particular assumption is made
on the velocity profile in the material layer. The models
are written in arbitrary coordinate system, and several
formulations are provided. A Coulomb friction term is derived
within the same framework, relevant in particular for debris
avalanches. All our models are invariant under rotation,
admit a conservative energy equation, and preserve the
steady state of a lake at rest.
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