Nearly Kähler manifolds with positive holomorphic sectional curvature
Two problems are studied for connected compact nearly Kähler manifolds \((M,g,J)\), i.e. almost Hermitian manifolds such that \((\nabla_ XJ)X=0\) for all vector fields \(X\). First, when \(M\) is an Einstein manifold, the authors derive an integral formula on the unit sphere bundle over M. Using this formula, they obtain several results concerning pinching problems for the holomorphic sectional curvature. Secondly they consider the following conjecture of A. Gray: Let \((M,g,J)\) be a compact nearly Kähler manifold with positive sectional curvature and constant scalar curvature. Then the manifold is isometric to complex projective space with a metric of constant holomorphic sectional curvature or a 6-dimensional sphere with a Riemannian metric of constant sectional curvature. It is known that the answer to this conjecture is positive for the class of Kähler manifolds. However, the answer is negative for the class of non-Kähler nearly Kähler manifolds. The counterexample constructed in this paper is the homogeneous 6-dimensional space \(Sp(2)/U(1)\times Sp(1)\). This space is diffeomorphic to a complex projective space of complex dimension 3 and it carries a metric and an almost Hermitian nearly Kähler (non-Kähler) structure with strictly positive but non-constant sectional curvature. It is also a Riemannian 3-symmetric space.
scientific article; zbMATH DE number 3915148
scientific article