Some submanifolds of a parakähler manifold (Q1389827)

An almost Hermitian manifold is called para-Kähler if its curvature tensor satisfies \[ R(X,Y,Z,W)-R(X,Y,JZ,JW)=R(JX,JY,Z,W) \] for any tangent vectors \(X,Y,Z,W\) where \(J\) is the almost complex structure. The purpose of this article is to study even-dimensional totally umbilical submanifolds \(M\) of a para-Kähler manifold. The authors derive some relations involving sectional and bisectional curvature of \(M\) when \(M\) is weakly anti-holomorphic. They also prove that if the submanifold \(M\) is para-Kähler and the ambient space is of constant sectional curvature \(c\), then \(M\) is flat and \(c\leq 0\). As an application, they show that a result of \textit{B.-Y. Chen} concerning extrinsic spheres of a Kähler manifold [Mich. Math. J. 23, 327-330 (1976; Zbl 0338.53043)] can be improved if one assumes that the extrinsic spheres are totally real submanifolds.