The concept of nearly Kähler manifold was defined and studied by A. Gray [J. Differ. Geom. 4, 283–309 (1970; Zbl 0201.54401)]. It refers to an almost complex Hermitian manifold \(M\equiv(M,I)\) satisfying \(\nabla_X(I)X=0\) for any vector field \(X\). Equivalently, \(\nabla\omega\) must be totally skew-symmetric, where \(\omega\) is the Hermitian form. If \(\nabla_X\omega\neq0\) for any vector field \(X\), then \(M\) is called strictly nearly Kähler. In [Asian J. Math. 6, No. 3, 481–504 (2002; Zbl 1041.53021)], P.-A. Nagy has shown that any strictly nearly Kähler manifold is locally the product of locally homogeneous manifolds, strictly nearly Kähler 6-manifolds, and twistor spaces of quaternionic Kähler manifolds of positive Ricci curvature, equipped with the Eels-Salamon metric. Thus the term “nearly Kähler” is nowadays used for strictly nearly Kähler 6-manifolds.
This paper begins by recalling several characterizations of nearly Kähler manifolds, examples, its relevance in geometry and physics, and its local structure, which gives rise to a bigrading of the de Rham graded algebra \(\Lambda(M)\) of complex valued differential forms. Then the decomposition of the de Rham differential map into bihomogeneous components is considered, \(d=N+\partial+\bar\partial+\bar N\) with corresponding bidegrees \((2,-1)\), \((1,0)\), \((0,1)\) and \((-1,2)\); in particular, \(N\) is given by the dual of the Nijenhuis operator. The main purpose of the paper is to establish certain relations among the Laplacians defined by \(d\) and its components, \(\Delta_d\), \(\Delta_N\), \(\Delta_\partial\), \(\Delta_{\bar\partial}\) and \(\Delta_{\bar N}\), generalizing some Kähler identities to nearly Kähler manifolds.
Those relations use the supercommutator of two graded homomorphisms of a graded commutative ring with unit, \(\{x,y\}=xy-(-1)^{\tilde x\tilde y}yx\), where \(\tilde x\) denotes the parity of \(x\), as well as the corresponding concept of algebraic differential operator of algebraic order \(i\); they are defined by induction on \(i\), like usual differential operators on smooth functions, by using \(\{\;,\;\}\) instead of the usual commutator. Some basic identities of this theory are recalled and applied to \(d\) and its components on \(\Lambda(M)\). In this way, \(\Delta_d=\{d,d^*\}\), \(\Delta_\partial=\{\partial,\partial^*\}\), etc., and the additional Laplacians \(\Delta_{\partial-\bar\partial}=\{\partial-\bar\partial,\partial^*-\bar\partial^*\}\) and \(\Delta_{N+\bar N}=\{N+\bar N,N^*+\bar N^*\}\) are also considered. It is shown that \(\Delta_\partial-\Delta_{\bar\partial}=R\), where \(R\) is a scalar operator acting on \((p,q)\)-forms as multiplication by \(\lambda^2(3-p-q)(p-q)\), and \(\lambda\) is a non-zero real constant determined by the equalities \(d\omega=3\lambda\,\text{Re}\,\Omega\) and \(d\,\text{Im}\,\Omega=-2\lambda\omega^2\) for some \((3,0)\)-form \(\Omega\) with \(|\Omega|=1\). It is also proved that \(\Delta_d=\Delta_{\partial-\bar\partial}+\Delta_{N+\bar N}=\Delta_\partial+\Delta_{\bar\partial}+\Delta_N+\Delta_{\bar N}-\{\partial^*,\bar\partial\}-\{\partial,\bar\partial^*\}\). It follows that a differential form is harmonic if and only if it is in the kernel of \(\Delta_\partial\), \(\Delta_{\bar\partial}\) and \(\Delta_{N+\bar N}\), simultaneously. Then the main result of the paper follows: the space of harmonic \(i\)-forms on \(M\) has a direct sum decomposition on harmonic forms of pure Hodge type, \(\mathcal{H}^i(M)=\bigoplus_{i=p+q}\mathcal{H}^{p,q}(M)\), where \(\mathcal{H}^{p,q}(M)=0\) unless \(p=q\), or \(p=2\) and \(q=1\), or \(q=1\) and \(p=2\).