Consider a surface $S$ smoothly embedded in $\mathbb{R}^3$. Classically, the (Riemannian) curvature of $S$ is described by the second fundamental form, which is constructed from partial derivatives of a local parameterization.

Alternatively, is there a "nice" variational characterization of surface curvature? (E.g., one that does not depend on local parameterizations but only on the metric $g$ induced by the embedding.) In other words, is there a scalar functional whose minimizer completely describes the Riemannian curvature?

One idea that comes to mind is that Riemannian curvature is the curvature associated with the Levi-Civita connection -- hence, you might try to construct a functional over the set of metric connections on $(S,g)$ that penalizes torsion.

(This question is motivated by discrete (e.g., piecewise linear or simplicial) differential geometry, where local differential quantities are ill-defined but metric quantities are available nonetheless.)

Consider a surface $S$ smoothly embedded in $\mathbb{R}^3$. Classically, the (Riemannian) curvature of $S$ is described by the second fundamental form, which is constructed from partial derivatives of a local parameterization.

Alternatively, is there a "nice" variational characterization of surface curvature? (E.g., one that does not depend on local parameterizations but only on the metric $g$.) In other words, is there a scalar functional whose minimizer completely describes the Riemannian curvature?

One idea that comes to mind is that Riemannian curvature is the curvature associated with the Levi-Civita connection -- hence, you might try to construct a functional over the set of metric connections on $(S,g)$ that penalizes torsion.

(This question is motivated by discrete (e.g., piecewise linear or simplicial) differential geometry, where local differential quantities are ill-defined but metric quantities are available nonetheless.)