The Gauss - Bonnet - Grotemeyer Theorem in spaces of constant curvature ∗

In 1963, K.P. Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R 3 with Euler characteristic χ(M), Gauss curvature G and unit normal vector field n. Grote-meyer's identity replaces the Gauss-Bonnet integrand G by the normal moment (a · n) 2 G, where a is a fixed unit vector: M (a · n) 2 Gdv = 2π 3 χ(M). We generalize Grotemeyer's result to oriented closed even-dimesional hypersurfaces of dimension n in an (n + 1)-dimensional space form N n+1 (k).