Riemannian almost-product structures with maximal mobility (Q1098116)

Let \(n_ 1,...,n_ r\) be positive integers, \(n=n_ 1+...+n_ r\). A Riemannian almost-product structure of type \((n_ 1,...,n_ r)\) is an n-dimensional manifold M together with an \(O(n_ 1)\times...\times O(n_ r)\)-structure on M. The almost product structure is said to have maximal mobility if its group G of automorphisms has maximal dimension \(1/2 \sum_{i}n_ i(n_ i+1)\). Such structures are called \({\mathcal M}\)- structures by \textit{H. Gauchman} [Tensor, New Ser. 31, 13-26 (1977; Zbl 0354.53030)], who gave a local classification of them, by using a certain class of Lie algebraic objects, the so-called M-objects of type (s,p,q). An M-object permits to write down the structural equations of the corresponding \({\mathcal M}\)-structure, and thus to obtain a natural one-to- one correspondence between the set of all locally irreducible \({\mathcal M}\)- structures and the set of all irreducible M-objects, of corresponding types. In the present paper the author gives a classification of local, or simply-connected global, \({\mathcal M}\)-structures up to isomorphism. In order to avoid the lengthy computations with tensor fields associated with \({\mathcal M}\)-structures, he regards an \({\mathcal M}\)-structure from the beginning as the homogeneous space of its automorphism group G. Transferring the data of the \({\mathcal M}\)-structure of the Lie algebra of G yields a so-called \({\mathcal L}\)-object. The algebraic structure of \({\mathcal L}\)-objects allows to reduce \({\mathcal L}\)-objects to \({\mathcal L}\) *-objects, which are essentially Gauchman's M-objects, but with some differences, because of certain inaccuracies in Gauchman's calculations, pointed out by the author (see pp. 109 and 119). The main theorem states that there are natural bijective correspondences between the isomorphism classes of irreducible \({\mathcal M}\)-structures, \({\mathcal L}\)-objects and \({\mathcal L}\) *- objects, of corresponding types. Thus, he obtains the desired complete classification. As an application of the general result and of his view of the problem in terms of homogeneous spaces, he determines all \({\mathcal M}\)-structures on simply-connected spaces of constant curvature.