标题： 次黎曼曲率与Heisenberg群中的Gauss-Bonnet定理 显示英文标题

标题： Sub-Riemannian curvature and a Gauss-Bonnet theorem in the Heisenberg group

摘要： 我们采用黎曼逼近方法，在Heisenberg群$\mathbb{H}$中定义了欧几里得$C^{2}$光滑曲面（远离特征点）的$\textit{sub-Riemannian Gaussian curvature}$概念，以及曲面上欧几里得$C^{2}$光滑曲线的$\textit{sub-Riemannian signed geodesic curvature}$概念。然后利用这些结果证明了Heisenberg群中的Gauss-Bonnet定理。此外，还给出了一个应用，即在$\mathbb{H}$中Carnot-Carathéodory距离的Steiner公式。 显示英文摘要

摘要： We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carath\'eodory distance in $\mathbb{H}$ is provided.