Geometry and symmetry on Sasakian manifolds

Abstract

In [26] the second author has given a brief survey about the local generalization to arbitrary Riemannian manifolds of the notion of a reflection with respect to a point, a line or a linear subspace in a Euclidean space. Local symmetries with respect to a point (local geodesic symmetries) are well-known and these local diffeomorphisms are already used at many occasions to study and even to classify some particular classes of Riemannian manifolds (see for example [25]). Local symmetries with respect to a curve, in particular with respect to a geodesic, led to less well-known characterizations of locally symmetric manifolds and spaces of constant curvature [27]. Finally, local symmetries with respect to submanifolds are studied in [21] and they give some nice geometrical resultsin the theory of embedded minimal and totally geodesic submanifolds. Local symmetries with respect to a curve may also be used to study contact geometry, in particular on Sasakian manifolds. On these manifolds, the integral curves of the characteristic vector field are geodesies and the study of the local symmetries with respect to these curves led in a natural way to a very geometrical treatment of the so-called ^-symmetric spaces introduced in [18], These spaces are natural analogues of locally symmetric spaces. (See [4], [5], [6], [18].) In this paper we continue our study of Sasakian geometry but now we focus on local symmetries with respect to geodesies which cut the integral curves of the characteristicvector fieldorthogonally. Such geodesies are usually called ^-geodesies. The main purpose is to use these local symmetries to give new characterizations of ^-symmetric spaces, Sasakian space forms and locally symmetric Sasakian manifolds. In sections 2, 3 and 4 we treat some general preliminaries about Sasakian manifolds and symmetries with respect to a curve thereby focussing on the central role of normal coordinates, Fermi coordinates and Jacobi vector fields. (For more details about contact geometry we refer to [1], [29].) In sections 5, 6 and 7 we prove our main results about symmetry.