# Curvature of a curve and curvature form

Is there any relation between the curvature of a curve defined in a classical way as the reciprocal of the radius of the osculating circle in a point and the curvature form on Riemannian manifolds?

That said, the Gaussian curvature at a point $p$ of a surface $S$ immersed in $\mathbf{R}^{3}$ can be computed intrinsically (and is equal to the sectional curvature of $S$ at $p$, possibly up to a mutiplicative factor depending on conventions), but is also the product of the principal curvatures of $S$ at $p$, and the latter may be interpreted as reciprocal radii of normal sections of $S$ through $p$.