论文标题
几何拉普拉斯法
A geometric Laplace method
论文作者
论文摘要
近似积分的经典工具是拉普拉斯方法。一阶以及高阶拉普拉斯公式最常用坐标书写而没有任何几何解释。在本文中，以最佳运输为动机，我们给出了Laplace方法的一阶术语的几何表述。中央工具是Kim-McCann Riemannian指标，该度量是在最佳运输领域引入的。我们的主要结果用标准的几何对象（例如音量形式，拉普拉斯主义者，协变量衍生物和标量曲率）在Kim-McCann框架中自然产生的两个不同指标的标曲率表达了一阶项。经过，我们给出了Laplace公式的明确量化版本以及应用程序的示例。
A classical tool for approximating integrals is the Laplace method. The first-order, as well as the higher-order Laplace formula is most often written in coordinates without any geometrical interpretation. In this article, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation of the first-order term of the Laplace method. The central tool is the Kim-McCann Riemannian metric which was introduced in the field of optimal transportation. Our main result expresses the first-order term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two different metrics arising naturally in the Kim-McCann framework. Passing by, we give an explicitly quantified version of the Laplace formula, as well as examples of applications.
