The differential geometry of almost Hermitian almost contact metric submersions

Three types of Riemannian submersions whose total space is an almost Hermitian almost contact metric manifold are studied. The study is focused on fundamental properties and the transference of structures. 1. Introduction. In this paper, we discuss some geometric properties of Riemannian submersions whose total space is an almost Hermitian almost contact metric manifold. If the base space is an almost quaternionic metric manifold, Watson has defined in [12, 13] a type of such submersions which we will call almost Hermitian almost contact metric submersion of type I. When the base space is an almost contact metric manifold with 3-structure, another type of these submersions called almost Hermitian almost contact metric submersions of type II has been introduced by the present author in [9]. Replacing the base space by an almost Hermitian almost contact metric manifold, we get a new type of such submersions, a third one, which we will call almost Hermitian almost contact metric submersions of type III. Note that this last type lies between almost Hermitian submersions studied by Watson [11] and almost contact metric submersions of type I [6, 8, 15]. Analogously, almost Hermitian almost contact metric submersions of type I lie between almost Hermitian submersions and almost contact metric submer-sions of type II [6, 8, 15]. This text is organized in the following way. Section 2 is devoted to the background of the manifolds which will be used in the sequel. Section 3 is concerned with the properties of the three types of submersions under consideration. For each type, we have here examined: (1) the structure of the base space and the fibres according to that of the total space; (2) the classes of submersions with totally geodesic fibres; (3) the classes of submersions preserving the holomorphic sectional curvature ten-sor of the vertical or of the horizontal vector fields. In Section 4, we give some examples of these types of submersions. Throughout this paper, arbitrary vector fields of the tangent space of a differentiable manifold M will be denoted by D, E, and G.