Grinberg, Eric L.; Li, Haizhong The Gauss-Bonnet-Grotemeyer theorem in space forms. (English) Zbl 1208.53059 Inverse Probl. Imaging 4, No. 4, 655-664 (2010). 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, 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 \(N^{n+1}(k)\) Reviewer: Constantin Călin (Iaşi) Cited in 1 Document MSC: 53C40 Global submanifolds 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature Keywords:Gauss-Bonnet theorem; Gauss-Kronecker curvature; hypersurfaces; Grotemeyer Citations:Zbl 0117.38401 PDFBibTeX XMLCite \textit{E. L. Grinberg} and \textit{H. Li}, Inverse Probl. Imaging 4, No. 4, 655--664 (2010; Zbl 1208.53059) Full Text: DOI