The Gauss-Bonnet-Grotemeyer Theorem in space forms

Eric L Grinberg,Haizhong Li

doi:10.3934/ipi.2010.4.655

Abstract

In 1963, K.P.~Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let $M$ be an oriented closed surface in theEuclidean space $\mathbb R^3$ with Euler characteristic $\chi(M)$, Gauss curvature $G$ and unit normal vector field $\vec n$. Grotemeyer'sidentity 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-dimensionalhypersurfaces of dimension $n$ in an $(n+1)$-dimensional space form $N^{n+1}(k)$.

Highlights

Proposition 2.6 Let M be an n-dimensional hypersurface in the (n+1)-dimensional space form N n+1(k)
Since M is a closed hypersurface in N n+1(k), the Gauss-Bonnet Theorem states in this case that volSn(1) 2 χ(M
For n even and m odd, we have for any fixed unit vector in the linear space Ln+1(k), the following expression for M qmGdv: qmGdv

Summary

Introduction

P. Grotemeyer proved the following interesting result: Theorem 1.1 ([7]) Let M be an oriented closed surface in 3-dimensional Euclidean space R3 with Gauss curvature G and a unit normal vector field n. Let x : M → N n+1(k) be an immersed n-dimensional oriented closed hypersurface in the (n + 1)-dimensional space form N n+1(k), with Euler characteristic χ(M ), Gauss-Kronecker curvature G and unit normal vector field n. Corollary 1.2 Let M be an oriented closed surface in the 3-dimensional space form N 3(k) with extrinsic curvature G and unit normal vector field n.

Results

Conclusion