curvaturesquared

Curvaturesquared is a generic term used in mathematics and theoretical physics to denote quantities obtained by squaring curvature measures. Its precise meaning depends on the context.

In the study of curves in Euclidean space, the curvature κ(s) is a nonnegative scalar function of arc length. Squaring yields κ(s)^2, which features in energy functionals that penalize bending, most notably the bending energy E = ∫ κ^2 ds for a curve or filament. Such functionals appear in the theory of elastic rods and in computer graphics for smooth curve design.

For surfaces, curvature is described by invariants such as the mean curvature H and the Gaussian curvature K. Squared curvature invariants H^2 and K^2 arise in geometric analysis and variational problems. The Willmore energy, a classical functional, is proportional to ∫ H^2 dA over a surface, emphasizing the role of curvature-squared terms in surface optimization.

In physics, curvature-squared terms appear in effective field theories and gravity. Lagrangians that include R^2, R_{ab}R^{ab}, or R_{abcd}R^{abcd} extend general relativity with higher-derivative corrections and can influence early-universe cosmology and quantum gravity. In four dimensions, the Weyl tensor C_{abcd}C^{abcd} can be squared, C_{abcd}C^{abcd}, which is conformally invariant up to a total derivative. The Gauss-Bonnet combination R_{abcd}R^{abcd} - 4 R_{ab}R^{ab} + R^2 is topological in four dimensions but contributes in higher dimensions.

Overall, curvaturesquared is a versatile label reflecting the ubiquity of quadratic curvature expressions across geometry and physics.