The Laplace-Beltrami operator in almost-Riemannian Geometry
The Laplace-Beltrami operator in almost-Riemannian Geometry
复制标题
近黎曼几何中的拉普拉斯-贝尔特拉米算子
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
C. Laurent
中科院分区：
文献类型：
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作者：
U. Boscain;C. Laurent
二维准黎曼结构是表面上的广义黎曼结构，其局部正交框架由生成可共线的向量场对的李括号给出。一般来说，奇异集是一个嵌入的一维流形，有三种类型的点：黎曼点，其中两个向量场线性无关；格鲁辛点，其中两个向量场共线，但它们的李括号不共线；以及切点，其中两个向量场和它们的李括号共线，并且通过多一个括号获得缺失方向。一般来说，切点是孤立的。在本文中，我们研究了这种结构上的 Laplace-Beltrami 算子。在没有切点的紧致可定向曲面的情况下，我们证明了Laplace-Beltrami算子本质上是自伴的并且具有离散谱。因此，这种结构中的量子粒子无法穿过奇点，热量也无法流过奇点。这是一个有趣的现象，因为当接近奇异集（即向量场变得共线）时，所有黎曼量都会爆炸，但测地线仍然定义良好，并且可以在没有奇点的情况下穿过奇异集。这种现象也出现在不等规的亚黎曼结构中，即生长向量取决于点。我们通过分析 Martinet 案例来证明这一事实。
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet case.