Local Gauss-Bonnet theorem

Let $\mathrm{\Delta }$ be a geodesic triangle on a surface (pseudo-Riemannian manifold) of dimension 2.

where $K$ is the Gaussian curvature.

This theorem says that the angular excess of such a triangle is simply the total curvature inside it.

There is a visual idea of the proof in @needham2021visual page 23:

based on the additivity of the angular excess.

It has a surprising global version: global Gauss-Bonnet theorem.