  We establish various Hardy inequalities involving the distance function from
submanifolds of Riemannian manifolds, where the natural weights are expressed
in terms of bounds of the mean curvature of the submanifold and sectional/Ricci
curvature of the ambient Riemannian manifold. Our approach is based on subtle
Heintze-Karcher-type Laplace comparisons of the distance function and on a
D'Ambrosio-Dipierro-type weak divergence formula for suitable vector fields,
providing Barbatis-Filippas-Tertikas-type Hardy inequalities in the curved
setting. Under very mild assumptions, we also establish the sharpness and
non-existence of extremal functions within the Hardy inequalities and -
depending on the geometry of the ambient manifold - their extensibility to
various function spaces. Several examples are provided by showing the
applicability of our approach; in particular, well-known Hardy inequalities
appear as limit cases of our new inequalities.
