Almost Kähler manifolds whose antiholomorphic sectional curvature is pointwise constant

Abstract

We prove that an almost Kähler manifold (M, g, J) with dim M ≥ 8 and pointwise constant antiholomorphic sectional curvature is a complex spaceform. -Introduction and preliminariesLet (M, g, J) be a 2n-dimensional almost Hermitian manifold. A 2-plane α in the tangent space T x M at a point x of M is antiholomorphic if it is orthogonal to Jα.The manifold (M, g, J) has pointwise constant antiholomorphic sectional curvature (p.c.a.s.c.) ν if, at any point x, the Riemannian sectional curvature ν(x) = K x (α) is independent on the choice of the antiholomorphic 2-plane α in T x M.If (g, J) is a Kähler structure, the previous condition means that (M, g, J) is a complex space-form, i.e. a Kähler manifold with constant holomorphic sectional curvature µ = 4ν ([2]). Moreover, the Riemannian curvature tensor R satisfies:ν being a constant function and π 1 , π 2 the tensor fields such that: