Cartan's and Gauss's equations and rigidity theorems for isometric embeddings in low Sobolev regularity

What the math gives to ML

The paper supplies a distributionally valid version of Cartan's structural equations for continuous, low-Sobolev orthonormal frames, and derives the Gauss determinant identity for rough isometric graphs. The transferable asset is not the classical smooth identity itself, but the weak formulation: curvature can be enforced through first-order coframe and connection equations, with residuals integrated against test functions rather than evaluated from unstable pointwise Hessian determinants. This suggests a geometric-consistency loss for neural implicit surfaces or metric-learning networks that remains meaningful under limited smoothness and can be paired with weak finite-element-style quadrature. The most promising experiment is to compare weak Cartan residuals against direct Hessian-determinant Gauss losses when fitting noisy or highly oscillatory isometric embeddings.