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 $2n \ geq 4$ is a complex space form iff for every $X$ at any point $p$ the operator $\lamda_{X,JX}$ has eigen vectors in the plane $X \wedge JX$