Gauss Bonnet Theorem

Question

The Gauss Bonnet theorem for a closed surface states
$\int_M K = 2 \pi \chi$
where $\chi$ is the Euler characteristic and $K$ is the Gaussian curvature
$K = k_1 k_2$,
which is the multiplication of the principal curvatures, meaning the minimum and maximum curvature, of the surface at the point of evaluation. Is there an intuitive way to understand why the Gauss Bonnet theorem requires the use of the Gaussian curvature, instead of a combination of any other curvatures on the surface?
For example, if I subdivide a surface into square-plaquettes, can I define a topological number using the curvature along the sides of the plaquettes instead?
I simply do not understand where the principal curvatures come from and whether they are avoidable, ie, whether there are alternative formulations of the Gauss Bonnet theorem that do not rely on them.

Answer 1

The historical definition of the Gaussian curvature at a point $p$ of a surface in $\mathbb R^3$ was in terms of the product of the minimum and maximum curvatures, etc. However, a more conceptual definition would be to define the Gaussian curvature as the determinant of the Weingarten map $W_p$, also known as the shape operator. Here $W_p$ is an endomorphism of the tangent plane to the surface at the point $p$. The determinant, unlike the trace, is an intrinsic invariant of the Riemannian metric (i.e., the first fundamental form) on the surface. The reason that the Gaussian curvature occurs in the Gauss-Bonnet theorem rather than some other combination is because the Gaussian curvature is the only intrinsic invariant at a point, to begin with.