Compact quadratics-manifolds

Abstract

The definition of a Riemannian regular s-manifold (M, g, s) is similar to that of a Riemannian symmetric space but without the condition that symmetries have order two. A regularity condition (trivially satisfied for symmetric spaces) is imposed on the composition of symmetries. Such manifolds are known to be homogeneous ([2], [4]) and classification problems reduce essentially to the study of automorphisms of Lie groups. In this connection, recent work of Wolf and Gray [7] is of fundamental importance for cases when the symmetries have finite order. A metrisable regular s-manifold (M, s) is defined by relaxing the unique choice of g in (M, g, s) to that of any g compatible with the s-structure. There is an obvious equivalence relation on the class of such manifolds, and we seek theorems which are valid up to this equivalence. For any (M, s) there is an associated tensor field S of type (1, 1) and (M, g, s) is symmetric if and only if S has linear minimal polynomial. We treat the case when S has a quadratic minimal polynomial; then (M, s) is called a quadratic s-manifold. Any such (M, s) admits an almost complex structure cb canonically associated with S; moreover, either all symmetries have order 3 or q~ is integrable and there exists a metric g for which (M, g, s) is a Riemannian regular s-manifold and (M, g) is Hermitian symmetric with respect to q~. This paper gives a classification up to equivalence of all compact quadratic (M, s). w 1 is mostly expository, but improves slightly some known results; it contains most basic definitions and properties for later use. In particular, it is easily seen that for any (M, s) the simply connected covering space ~ of M admits a metrisable structure (/~t, g) whose symmetries cover those of (M, s). Then if (M, s) and (m ' , s') are covered by (A~t, g) the equivalence of (M, s) and (M', s ') reduces to a study of certain deck transformations of A~. We also develop for later use the relation between (M, s) and a triple (G, H, 0) where G is a Lie group acting transitively on M with isotropy group H, and 0 is an automorphism of G determined by s. The section concludes with some remarks on the smoothness of the map s and tensor field S. In w the notion of a quadratic s-manifold is defined and four theorems are stated