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

Let $\{\eta^i\}_{i=1}^2$ be a an orthonormal coframe on a domain $U$ on a smooth surface $(\Sigma,g)$. When $\eta^i$ is smooth, it is well-known that there is a unique connection 1-form $\omega$ verifying Cartan's first structural equations $d\eta^i = (*\eta^i) \wedge \omega$, and Cartan's second structural equation $d\omega = K_g dvol_g$. We prove that this statement remains valid when the frame is $C^0 \cap H^{\frac12}$, where the structural equations are understood in the sense of distributions. From this, we deduce that the Gauss equation $\mathrm{Det}\, D^2 f = K_g (1+|Df|^2)^2$ holds for every graphical representation $f$ of an isometric embedding of regularity $C^1 \cap W^{1+\frac23,3}$ or $c^{1,\frac12} \cap BV^2$. As an application, we prove regularity and convexity results for isometric embeddings of closed surfaces and convex caps with $K_g \geq 0$.