Integral Geometry and the Gauss-bonnet Theorem in Constant Curvature Spaces
نویسنده
چکیده
We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.
منابع مشابه
The Gauss - Bonnet - Grotemeyer Theorem in spaces of constant curvature ∗
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 R 3 with Euler characteristic χ(M), Gauss curvature G and unit normal vector field n. Grote-meyer's identity replaces the Gauss-Bonnet integrand G by the normal moment (a · n) 2 G, where a is a fixed unit vector: M (a · n) 2 Gdv = 2π 3 χ(M). We generaliz...
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