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The Gauss-Bonnet-Grotemeyer theorem in space forms. (English) Zbl 1208.53059

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 \(N^{n+1}(k)\)

MSC:

53C40 Global submanifolds
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 0117.38401
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