  This paper is devoted to Hardy inequalities concerning distance functions
from submanifolds of arbitrary codimensions in the Riemannian setting. On a
Riemannian manifold with non-negative curvature, we establish several sharp
weighted Hardy inequalities in the cases when the submanifold is compact as
well as non-compact. In particular, these inequalities remain valid even if the
ambient manifold is compact, in which case we find an optimal space of smooth
functions to study Hardy inequalities. Further examples are also provided. Our
results complement in several aspects those obtained recently in the Euclidean
and Riemannian settings.
