The Gauss-Bonnet-Grotemeyer theorem in space forms (Q618999)

Let \(M\) be an oriented closed surface in the \(3\)-dimensional Euclidean space \(\mathbb R^3\) with Gauss curvature \(G\) and unit vector field \({\vec{n}}\). By replacing the Gauss curvature \(G\), with \(({\vec{a}\cdot {\vec{n}}})^2G\), where \({\vec{a}}\) is a fixed unit vector field, \textit{K. P. Grotemeyer} extended the well-known Gauss-Bonnet theorem [Ann. Acad. Sci. Fenn., Ser. A I 336, No. 15 (1963; Zbl 0117.38401)]. In this paper, the authors generalize Grotemeyer's result to oriented closed even-dimensional hypersurfaces of dimension \(n\) in an \((n+1)\)-dimensional space form \(N^{n+1}(k)\)