论文标题
riemannian近似方案中表面的高斯和正常曲率的极限
Limit of Gaussian and normal curvatures of surfaces in Riemannian approximation scheme for sub-Riemannian three dimensional manifolds and Gauss-Bonnet theorem
论文作者
论文摘要
在Heisenberg Group中，作者在Arxiv中的Balogh-Tyson-Vecchi：1604.00180使用Riemannian近似方案$（\ Mathbb h^1，<，> _ l）$，在Gromov引入的Heisenberg Group中$ l \ rightarrow \ infty $。他们表明，这些限制存在（与Riemannian表面积或长度形式的极限不同），并且它们以$ \ Mathbb h^1 $为$（\ Mathbb h^1，<，> _ l）$ L $时，它们以$ \ Mathbb h^1 $的限制获得Gauss-Bonnet定理。 Wang-Wei在Arxiv：1912.00302中扩展到了Aggine Group和Minkowski平面的刚性运动。我们将这两篇论文的构造概括为亚riemannian三维歧管中的表面，遵循Arxiv的方法：1909.13341，并证明了类似的高斯 - 骨网定理。
The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme $(\mathbb H^1,<,>_L)$, in the Heisenberg group, introduced by Gromov, to calculate the limits of Gaussian and normal curvatures defined on surfaces of $\mathbb H^1$ when $L\rightarrow\infty$. They show that these limits exist (unlike the limit of Riemannian surface area form or length form), and they obtain Gauss-Bonnet theorem in $\mathbb H^1$ as limit of Gauss-Bonnet theorems in $(\mathbb H^1,<,>_L)$ when $L$ goes to infinity. This construction was extended by Wang-Wei in arXiv:1912.00302 to the affine group and the group of rigid motions of the Minkowski plane. We generalize constructions of both papers to surfaces in sub-Riemannian three dimensional manifolds following the approach of arXiv:1909.13341, and prove analogous Gauss-Bonnet theorem.
