A classical tool for approximating integrals is the Laplace method. The first-
and higher-order Laplace formulas are 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 to
the first-order term of the Laplace method. The central tool is a metric
introduced by Kim and McCann 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 metrics arising naturally in the Kim–McCann framework. We give an
explicitly quantified version of the Laplace formula, as well as examples of
applications.
