Symplectic reflections and complex space forms (Q915157)

Let (M,g,J) be a Hermitian manifold and N a submanifold of M. Then the reflection about N is called a symplectic reflection if it preserves the Kaehler form of (M,g,J). In this paper the authors prove the following Theorem. Let (M,g,J) be a Hermitian space of complex dimension \(n\geq 2\). Then it is a complex space form if and only if the local reflection with respect to any holomorphic surface is symplectic. This result generalizes Theorem 22 of [the authors, Geom. Dedicata 29, No.3, 259-277 (1989; Zbl 0673.53035)].