Cartan-type criterions for constancy of almost Hermitian manifolds

Abstract

We studied the axiom of anti-invariant 2-spheres and the axiom of co-holomorphic $(2n+1)$-spheres. We proved that a nearly K\"{a}hlerian manifold satisfying the axiom of anti-invariant 2-spheres is a space of constant holomorphic sectional curvature. We also showed that an almost Hermitian manifold $M$ of dimension $2m\geq6$ satisfying the axiom of co-holomorphic $(2n+1)$-spheres for some $n,$ where $(1\leq n\leq m-1)$, the manifold $M$ has pointwise constant type $\alpha$ if and only if $M$ has pointwise constant anti-holomorphic sectional curvature $\alpha$.