J-harmonic functions on almost Hermitian manifolds

Abstract

We define an operator ΔJ on any almost Hermitian manifold by first projecting the gradient of a function to the tangent space of its level set, and then taking the divergence. We call ΔJ the J-Laplacian operator. It is then shown that J-harmonic functions, which are functions H satisfying ΔJH=0, are first integrals of the vector field δω# where ω is the fundamental 2-form. This generalizes the notion of Hamiltonian functions on symplectic manifolds, and provides a refreshing way to interpret the seminal classification of almost Hermitian manifolds by Gray and Hervella. Existence and non-existence of J-harmonic functions under some conditions are explored, where a connection to the problem of existence of closed orbits in dynamical systems is revealed.