# Osculating circle and principal curvatures in Riemannian geometry

For a hypersurface $M$ of $\mathbb R^n$, there are relations between the principal curvatures $\kappa_i$ (eigenvalues of the second fundamental form $II$) and the radius $r_i$ of the osculating circles in the directions of eigenvectors of $II$ at a given point on $M$, i.e. $$r_i=\frac{1}{\kappa_i}.$$ However when $M$ is a hypersurface of a Riemannian manifold $N$, I wonder if we have similar relationship. More precisely, Let $p\in M$ and $\nu_p$ the normal direction of $M$ at $p$. Let $u\in T_pM$ be an eigenvector of $II_p$ and $\kappa$ corresponding eigenvalue. Consider $\exp_p:\langle \nu_p,u\rangle\subset T_pN\to N$, where $V:=\langle\nu_p,u\rangle$ is the subspace of $T_pN$ spanned by $u$ and $\nu_p$. Then i) does there exist a circle in $\exp_p(V)$ tangent to curve $\gamma \subset \exp_p(V)\cap M$ such that its curvature at $p$ is $\kappa$? ii) is there any relation between $\kappa$ and the radius of this circle?