黎曼单形与三角剖分
Riemannian simplices and triangulations
摘要：
研究了任意维数$n$的黎曼流形中几何单形的一个自然的内禀定义，并利用这些单形得到了紧致黎曼流形的三角剖分准则.这些几何单形是使用Karcher平均来定义的。给定流形上一个凸集中的一个有限集顶点，使到顶点的距离平方的加权和最小的点是相对于权重的Karcher平均。利用重心坐标作为权值，得到了从标准欧氏单形到流形的光滑映射。一个黎曼单形被定义为这个重心坐标映射的像。在这项工作中，我们阐明的标准，保证重心坐标映射是一个光滑的嵌入。如果不是，我们说黎曼单形是退化的。欧几里得单形的“厚度”或“肥胖”的质量度量可以适用于这些黎曼单形。对于2维流形，单形是非退化的，如果它有一个正的质量测度，就像在欧几里德的情况下。然而，当维度大于2时，只有当质量超过取决于单形的大小和流形的截面曲率的绝对值的局部边界的正边界时，才能保证非退化。对非退化黎曼单形的几何分析得出了保证单纯复形同胚于流形的条件。
英文摘要：
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension $n$, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the "thickness" or "fatness" of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.
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