No abstract

Using the formalism of complex ℋ-spaces, we show that all real, Euclidean self-dual spaces that admit (at least) one Killing vector may be gauged so that only two distinct types of Killing vectors appear; in Kähler coordinates these are the generators of a translational or a rotational symmetry. We give explicit forms both for the Killing vectors and for the constraint on the Kähler potential function Ω which allows for such a Killing vector. In the translational case we show how all such spaces are determined by the general solution of the three-dimensional, flat Laplace’s equation and how these are related to the multi-Taub–NUT metrics of Gibbons and Hawking. In the rotational case we simplify the equation determining Ω, but this is not sufficient to obtain the general solution.

A Sasakian structure S=(ξ,η,Φ,g) on a manifold M is called positive if its basic first Chern class c 1 (F ξ ) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This allows us by example to give a completely independent proof of a result of Sha and Yang [SY] that for every positive integer k the 5-manifolds k#(S 2 ×S 3 ) admit metrics of positive Ricci curvature.

Introduction. In 1960 Sasaki [17] introduced a geometric structure related to an almost contact structure on a smooth manifold. This structure, which became known as a Sasakian structure, was studied extensively in the 1960's by an entire school of Japanese geometers (See [24] and references therein). In 1970 Kuo [14] refined this notion and introduced manifolds with Sasakian 3-structures. The same year Kuo and Tachibana, Tachibana and Yu, and Tanno [15,22,21] published foundational papers discussing Sasaklan 3-structures an'd these structues were then vigorously studied by many Japanese mathematicians from 1970-1975. This intense analysis culminated with an important paper of Konishi [13] which shows the existence of a Sasakian 3-structure on a certain principal SO(3) bundle over any quaternionic Khler manifold of positive scalar curvature. Earlier on, in 1973, Ishihara [10] had shown that if the distribution formed by the three Killing vector fields which define the Sasakian 3-structure is regular then the space of leaves is a quaternionic Kfihler manifold. This fact led Ishihara to his foundational work on quaternionic Kihler manifolds [9]. Ishihara's and Konishi's observation that quaternionic Khler and 3-Sasakian geometries are related is fundamental.

Let $M$ be a closed manifold of Sasaki type. A polarization of $M$ is defined by a Reeb vector field, and for one such, we consider the set of all Sasakian metrics compatible with it. On this space, we study the functional given by the squared $L^2$-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open.Comment: 36 pages, minor corrections made, example adde