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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?

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Briefly, no: Intrinsically, every real curve is flat. Curvature in the Riemannian sense is a measure of non-commutativity of covariant derivatives. Loosely, on a curve there's "only one direction", i.e., "not enough room" for covariant derivatives to fail to commute.

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 $p$.

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