The aim of the paper is to give a classification up to isomorphism of (local or simply-connected global) Riemannian almost-product structures (i.e. O(nl)x ... x O(nr)structures) whose automorphism group has maximal dimension (the so-called Jg-structures). The describing objects are Lie algebras with some additional structure, and the methods are mainly Lie algebraic. The results obtained are applied to determine all ~-structures on simply-connected spaces of constant curvature.
O. INTRODUCTION
An almost-product structure of type (n~ .... ,n,) on an n-dimensional Riemannian manifold M (n=n 1 +... +nr) consists of r pairwise orthogonal distributions D~,...,D r on M of dimensions n~ .... ,nr, respectively. Its group of automorphisms is a Lie group of dimension at most Β½Zini(ni + 1). Naturally, almost-product structures which reach this bound attract a particular attention, being the objects with highest symmetry. They have been called almost-product structures with maximal mobility, or briefly o///-Structures, in [2], a terminology which we adopt here. In that paper, H. Gauchman described (local) ~/-structures by means of certain algebraic objects. But his paper seems to contain some errors, and thus the classification remains incomplete. Besides we feel that the approach to the problem chosen there is not very enlightening, since it involves extensive calculations in local coordinates. There is also the paper Ill of E. H. Cattani and L. N. Mann dealing with (global) .~//-structures, but these authors stick to special cases and do not investigate ~/-structures of general type.
In this note we try to give a complete classification of (local, or simplyconnected global) ~-structures up to isomorphism. The describing objects -Lie algebras with some additional structure, we call them f *-objectsare essentially the same as Gauchman's M-objects, the differences being due to the inaccuracies mentioned above. But rather than entering into the lengthy and tedious calculations with tensor fields associated with .~/structures we regard an ~/-structure from the beginning as the homogeneous space of its automorphism group G. Transferring the data of the A-structure into the Lie algebra of G yields what we call an f-object. Since the .~/-structure can be recovered from the f-object, both concepts are equivalent (Section 2). In Section 3 we detect the algebraic structure of f-objects which allows us to reduce f-objects to f*-objects (Section 4). In this way the proof is confined to very elementary Lie algebraic