We define an operator ${\mathrm{\Delta }}_J$ 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 ${\mathrm{\Delta }}_J$ the J-Laplacian operator. It is then shown that J-harmonic functions, which are functions H satisfying ${\mathrm{\Delta }}_{J}H=0$, are first integrals of the vector field $\delta {\omega }^{\mathrm{\#}}$ 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.

我们定义一个运算符 ${\mathrm{\Delta }}_{Ĵ}$首先将函数的梯度投影到其水平集的切线空间，然后求散度，然后在几乎所有的厄米流形上进行求解。我们称之为${\mathrm{\Delta }}_{Ĵ}$所述Ĵ -Laplacian操作者。然后，示出了Ĵ K谐波的功能，其是功能ħ满足${\mathrm{\Delta }}_{Ĵ}H=0$是向量场的第一积分 $\delta {\omega }^{\mathrm{＃}}$其中ω是基本2形式。这概括了辛流形上哈密顿函数的概念，并提供了一种新颖的方式来解释Gray和Hervella对几乎埃尔米特流形的开创性分类。探索了在某些条件下J谐波函数的存在与不存在，揭示了与动力学系统中封闭轨道存在问题的联系。