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.

翻译：近似构件的经典工具是拉帕特法。第一阶以及高阶拉帕特公式通常在坐标上写成,没有几何解释。在本条中,由于在最佳运输方面出现的情况,我们给出了拉帕特法第一阶术语的几何公式。中心工具是金-麦肯里伊曼尼指标,这是在最佳运输领域引入的。我们的主要结果表现为第一阶术语,标准几何物体,如体积表、拉普拉西、可变衍生物和金-麦肯框架自然产生的两种不同度量的卡路里曲线。经过时,我们给出了拉帕特公式的明确量化版本以及应用实例。