The Gauss formulas for Laplacians on submanifolds

TL;DR

This paper extends Gauss formulas to various Laplacians on submanifolds, providing new relations between intrinsic and extrinsic geometric operators with applications to fluid dynamics equations.

Contribution

It introduces generalized Gauss formulas for different Laplacians on submanifolds of any codimension, including Ricci and divergence formulas, enhancing geometric analysis tools.

Findings

Derived Gauss formula for Ricci operator

Formulas for divergence of second fundamental form

Laplacian of 1-form on surfaces of revolution

Abstract

There are several types of Laplacians of a vector field on a Riemannian manifold. These include the Bochner and the Hodge Laplacian. The Gauss formula for the Levi-Civita connection relates the extrinsic connection to the intrinsic connection. We extend the Gauss formula for the connection to formulas for the different types of Laplacians of a vector field on a submanifold of any codimension $k\geq 1$. In the process, we derive a Gauss formula for the Ricci operator, formulas for the divergence of the second fundamental form, and a formula for the Laplacian of a $1$-form on a surface of revolution in terms of the Lie derivatives. The formulas have applications to the study of the formulation of the incompressible Navier-Stokes equations on a Riemannian manifold.