I've read many times that moving coframes where a convenient tool for computations in Riemannian Geometry, especially on surfaces, but never really used it. Lately to get a better feel of the method, I've tried to reprove classical results on surfaces using moving coframes and the Cartan structure equations :

Cartan's equations for surfaces :if $(\eta^1,\eta^2)$ is a orthonormal coframe on $(\Sigma^2,g)$, then $d\eta^1=\omega\wedge\eta^2$ and $d\eta^2=-\omega\wedge\eta^1$ for a unique $1$-form $\omega$, which moreover satisfies $d\omega=-K_g\eta^1\wedge\eta^2$.

What is really nice is that in these three equations we define at the same time the Levi-Civita connection (through $\omega$ which is the connection $1$-form) and the curvature, without ever actually computing covariant derivative.

I have been amazed at how some result get much easier in this framework, like the fact that $K_g=0$ implies local isometry to the euclidean metric, and also at how curvature computation become more straight forward.

Moreover a lot of the objects of interest in geometric analysis can be defined solely in terms of $1$-forms, the Hodge star $\ast$, exterior differential of forms and wedge product :

- $g(\nabla u,\nabla v)=\ast(du\wedge \ast dv)$.
- $\Delta u=\ast\ d\ast d u$.

As a geometric analyst, my favorite equation is probably Bochner's formula, which in the case of surfaces reads as : $$\tfrac{1}{2}\Delta |\nabla u|^2=\langle\nabla u,\nabla\Delta u\rangle+|\mathrm{Hess}\, u|^2+K_g|\nabla u|^2.$$ The usual proof of the Bochner formula involves commutations of covariant derivatives. My question is :

Question :Is it possible to prove the Bochner formula for surfaces relying on Cartan's structure equations instead of commutation of covariant derivative ?

It basically boils down to two subquestions :

- How does one compute $d\ast d\left(\ast(du\wedge \ast du)\right)$ ?
- Can one express $|\mathrm{Hess}\, u|^2$ in terms of the Hodge star, the connection form and exterior derivatives ?