A characterization of the complex space forms

Abstract

In the almost Hermitian geometry together with the classical Jacobi operator ${\lambda }_X$ we define also the linear symmetric operator ${\lambda }_{X,JX}$ where $X$ is a tangent vector at point $p\in M$. Then we prove the following theorem: A Kaehlerian manifold of dimension $2ngeq4$ is a complex space form iff for every $X$ at any point $p$ the operator ${\text{lamda}}_{X,JX}$ has eigen vectors in the plane $X\wedge JX$