In this paper we obtain a tangency principle for
hypersurfaces, with not necessarily constant $r$-mean
curvature function $H_r $, of an arbitrary Riemannian
manifold. That is, we obtain sufficient geometric conditions
for two submanifolds of a Riemannian manifold to coincide, as
a set, in a neighborhood of a tangency point. As applications
of our tangency principle, we obtain, under certain conditions
on the function $H_r$, sharp estimates on the size of the
greatest ball that fits inside a connected compact
hypersurface embedded in a space form of constant sectional
curvature $c\leq 0$ and on the size of the smallest ball that
encloses the image of an immersion of a compact Riemannian
manifold into a Riemannian manifold with sectional curvatures
limited from above. This generalizes results of Koutroufiotis,
Coghlan-Itokawa, Pui-Fai Leung, Vlachos and Markvorsen. We
also generalize a result of Serrin. Our techniques permit us
to extend results of Hounie-Leite.