Let me first recall what Gauss's Theorema Egregium says. Consider a surface isometrically embedded in $\mathbb{R}^3$. In some local coordinates, let the first and second fundamental forms be $$E \text{dx}^2 + 2F \text{dx} \text{dy} + G \text{dy}^2$$ and $$L \text{dx}^2 + 2M \text{dx} \text{dy} + N \text{dy}^2.$$ The Gaussian curvature is $$K=\frac{LN-M^2}{EG-F^2}.$$ Gauss's theorem says that despite this formula, $K$ only depends on the first fundamental form.

The proof of this basically algebraic, and comes down to some remarkable formulas (the Gauss Equations) arising from the equality of iterated mixed partial derivatives.

**Question**: What is the right algebraic setting for this? The only relationships between the first and second fundamental forms are the Codazzi–Mainardi equations, so it must be that somehow these equations force $K$ to be independent of the second fundamental form. These equations involve some partial derivatives, but given that no real analysis goes into Gauss's theorem I assume there must be a more general algebraic setting for this sort of result (but I have no idea what buzzwords to search for).