The Gauss-Bonnet-Grotemeyer Theorem in space forms

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 ℝ3 with Euler characteristic χ(M), Gauss curvature G and unit normal vector field n→. Grotemeyer's identity replaces the Gauss-Bonnet integrand G by the normal moment (a→ · n→)2G, where a is a fixed unit vector: ∫M (a→ · n→)2Gdv = 2π/3 χ(M). We generalize Grotemeyer's result to oriented closed even-dimensional hypersurfaces of dimension n in an (n + 1)-dimensional space form Nn+1(k). © 2010 American Institute of Mathematical Sciences.