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It is easy to see that curvature determines a plane curve up to rigid motions. For space curves, you need two quantities: the curvature and the torsion.

The latter can be observed from the Frenet-Serret equations $$\left[\begin{array}{c}T'\\N'\\B'\end{array}\right] = \left[\omega\right]_\times \left[\begin{array}{c}T\\N\\B\end{array}\right]$$ where $[\omega]_{\times}\in\mathfrak{so}(3)$ has three degrees of freedom; however, one of these is redundant (my intuition for this is that the twist of the normal and binormal about the tangent is irrelevant to describing the shape of the curve.)

For a surface in 3D, the second fundamental form is enough to determine the surface embedding up to rigid motions; but surely not all three distinct entries of the second fundamental form are needed? Can one recover the embedding (again, up to rigid motions) knowing only the mean curvature, for instance?

In general, given a $d$-dimensional Riemannian manifold, how many "curvatures" must be known at each point on the manifold to uniquely determine an embedding in $\mathbb{R}^n$ up to rigid motions (isometries of the ambient space)?

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