ar X iv : h ep - t h / 98 10 25 0 v 1 3 0 O ct 1 99 8 3 - Sasakian Manifolds

We begin this review with a brief history of the subject for our exposition shall have little to do with the chronology. In 1960 Sasaki [Sas 1] introduced a geometric structure related to an almost contact structure. This geometry became known as Sasakian geometry and has been studied extensively ever since. In 1970 Kuo [Kuo] refined this notion and introduced manifolds with Sasakian 3-structures (see also [Kuo-Tach, Tach-Yu]). Independently, the same concept was invented by Udri¸ste [Ud]. Between 1970 and 1975 this new kind of geometry was investigated almost exclusively by a group of and Yu. Already in [Kuo] we learn that the 3-Sasakian geometry has some interesting topological implications. Using earlier results of Tachibana about the harmonic forms on compact Sasakian spaces [Tach], Kuo showed that odd Betti numbers up to the middle dimension must be divisible by 4. In 1971 Kashiwada observed that every 3-Sasakian manifold is Einstein with a positive Einstein constant [Kas]. In the same year Tanno proved an interesting theorem about the structure of the isometry group of every 3-Sasakian space [Tan 1]. In a related paper he studied a natural 3-dimensional foliation on such spaces showing that, if the foliation is regular, then the space of leaves is an Einstein manifold of positive scalar curvature [Tan 2]. Tanno clearly points to the importance of the analogy with the quaternionic Hopf fibration S 3 → S 7 → S 4 , but does not go any further. In fact, Kashiwada's paper mentions a conjecture speculating that every 3-Sasakian manifold is of constant curvature [Kas]. She attributed this conjecture to Tanno and, at the time, these were the only known examples. Very soon after, however, it became clear that such a conjecture could not possibly be true. This is due to a couple of papers by Ishihara and Konishi [I-Kon, Ish 1]. They made a fundamental observation that the space of leaves of the natural 3-dimensional foliations mentioned above has a " quaternionic structure " , part of which is the Einstein metric discovered by Tanno. This led Ishihara to an independent study of this " sister geometry " : quaternionic Kähler manifolds [Ish 2]. His paper is very well-known and is almost always cited as the source of the explicit coordinate description of quaternionic Kähler geometry. Among other results Ishihara showed that his definition implies that the holonomy group of the metric is a subgroup …