论文标题
三维触点亚河畔歧管的表面上的随机过程
Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds
论文作者
论文摘要
我们关注三维触点亚riemannian歧管中表面上的随机过程。利用riemannian近似临近，利用Reeb矢量场，我们在表面上获得了二阶部分差分算子，该二级差分算子是Laplace-Beltrami操作员的极限。与限制性操作员相关的随机过程沿着接触分布在表面引起的特征叶面移动。我们表明，对于这种随机过程，椭圆特性点是无法访问的，而双曲线特征点可以从分离中访问。我们用示例说明了结果，并确定了海森伯格组中的规范表面，并以$ {\ rm su}（2）$和$ {\ rm sl}（2，\ mathbb {r}）$配备了标准的次级利曼式接触结构作为此设置的模型案例。我们的技术进一步使我们能够在一般的三维触点亚riemannian歧管中得出表面固有的高斯曲率的表达。
We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. Employing the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace-Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate the results with examples and we identify canonical surfaces in the Heisenberg group, and in ${\rm SU}(2)$ and ${\rm SL}(2,\mathbb{R})$ equipped with the standard sub-Riemannian contact structures as model cases for this setting. Our techniques further allow us to derive an expression for an intrinsic Gaussian curvature of a surface in a general three-dimensional contact sub-Riemannian manifold.
