Vorticity Gramian of compact Riemannian manifolds

TL;DR

This paper extends the concept of vorticity to n-dimensional compact Riemannian manifolds, analyzing its properties and establishing Helmholtz decomposition and Stokes' identity in this setting.

Contribution

It introduces a generalized vorticity concept for Riemannian manifolds and proves fundamental properties like Helmholtz decomposition and Stokes' identity.

Findings

Unique Helmholtz decomposition for vector fields on compact Riemannian manifolds

Stokes' type identity for smooth vector fields near the manifold

Extension of vorticity concept to higher-dimensional Riemannian geometry

Abstract

The vorticity of a vector field on 3-dimensional Euclidean space is usually given by the curl of the vector field. In this paper, we extend this concept to n-dimensional compact and oriented Riemannian manifold. We analyse many properties of this operation. We prove that a vector field on a compact Riemannian manifold admits a unique Helmholtz decomposition and establish that every smooth vector field on an open neighbourhood of a compact Riemannian manifold admits a Stokes' type identity.