Symplectic reflections and complex space forms

Abstract

We prove that a Hermitian manifold is a complex space form if and only if the local reflections with respect to any holomorphic surface are symplectic, i.e., preserve the Kihler form.

Divergence-preserving geodesic transformations

“…Since Q" is a volume form, symplectic geodesic transformations are volume-preserving. We refer to [4,5] for more information about symplectic reflections. Now, we shall prove that these last ones are the only possible examples.…”

Geometric properties of generalized symmetric spaces

“…The study of D'Atri spaces [23] is just a significant example, where the geodesic symmetries are assumed to be volume preserving. Within this general problem, other kinds of geodesic reflections (with respect to higher-dimensional submanifolds) were also investigated [20,21,42] as well as other kinds of geodesic transformations generalizing the similarities and inversions of the Euclidean space (see [22,30]).…”

Geodesic reflections in semi-Riemannian geometry

“…Considering the first condition in (7) in an analogous way as in [CV2], we obtain that (V^J)£ must be tangent to the submanifold. Now, M being of complex dimension greater than two and q > 1, there exists a timelike holomorphic surface normal to P and to {£,J£}.…”