Affine Connections and Gauss-Bonnet Theorems in the Heisenberg Group

Heisenberg群上的仿射联络和Gauss-Bonnet定理

In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean C2-smooth curves on surfaces.We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.

本文计算了Heisenberg群中远离示性点的欧几里得C²光滑曲面对应于两种Schou-ten-VanKampen仿射联络和容许联络的高斯曲率的次黎曼极限,以及曲面上欧几里得C2光滑曲线的对应于两种Schouten-VanKampen仿射联络和容许联络的符号测地曲率,得到了Heisen-berg群中对应于两种Schouten-VanKampen仿射联络和容许联络的Gauss-Bonnet定理.