Scope of classical surface Gauss-Bonnet

The result proved and used on this page is the two-dimensional Gauss-Bonnet identity Global Gauss-Bonnet for closed oriented surfaces, ${\int }_{M}KdA=2\pi \chi (M)$ for a closed oriented Riemannian surface, together with its boundary forms: it equates a curvature integral with an integer topological invariant.

It is a curvature/Euler-characteristic identity, not a classification theorem. Passing from the curvature identity to a genus formula, a normal form, or a homeomorphism type requires additional topological results. The topological classification of compact connected surfaces supplies one such route and is developed separately on the page classification-of-compact-connected-surfaces and is not proved or used here. Formulas such as $\chi =2-2g$ for orientable genus-$g$ surfaces are consequences of that classification, though classification is not the only way to establish a genus formula; they are not inferred from the curvature identity on this page.

Likewise this page proves nothing about higher dimensions. The Chern-Gauss-Bonnet theorem identifies, in even dimensions, the integral of a characteristic form built from the curvature (the Pfaffian or Euler form) with the Euler characteristic, and it belongs to characteristic-class and Chern-Weil theory; no Pfaffian form, characteristic class, transgression or connection on a general vector bundle is constructed here, and the higher-dimensional theorem is neither proved nor used. Nor is a Poincare-Hopf statement asserted. The surface theorem is self-contained on this page, and no consequence of the higher-dimensional theory is imported back into it.