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The Gauss-Bonnet-Grotemeyer Theorem in space forms

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  • 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 $\mathbb R^3$ with Euler characteristic $\chi(M)$, Gauss curvature $G$ and unit normal vector field $\vec n$. Grotemeyer's identity replaces the Gauss-Bonnet integrand $G$ by the normal moment $ ( \vec a \cdot \vec n )^2G$, where $a$ is a fixed unit vector: $ \int_M(\vec a\cdot \vec n)^2 Gdv=\frac{2 \pi}{3}\chi(M) $. We generalize Grotemeyer's result to oriented closed even-dimensional hypersurfaces of dimension $n$ in an $(n+1)$-dimensional space form $N^{n+1}(k)$.
    Mathematics Subject Classification: Primary: 53C42; Secondary 53A10.


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