We study Ambrose-Singer connections with an algebraic curvature tensor on simply connected manifolds carrying a homogeneous Riemannian structure of class ~ in the classification given by F. Tricerri and L. Vanhecke.
0. Introduction
Let (M,g) be a simply connected, n-dimensional manifold admitting a homogeneous Riemannian structure of class f3-Thus, there exists an alternating tensor field T of type (1, 2) satisfying the equations of Ambrose and Singer listed as (AS) (i), (ii), (iii) in [8] and condition (iv) in [8].
We denote by V the Riemannian connection with curvature tensor R and by V = V -T the related Ambrose-Singer connection with curvature/~ and torsion Y, = -2T.
For notation and general properties we refer the reader to [8], [6], [7], [11].
We recall that a simply connected manifold (M, g) admits a homogeneous Riemannian structure T 6 J3 if and only if it is a naturally reductive space [11]. Furthermore, such manifolds have been completely classified in dimension n ~< 5 by O. Kowalski, F. Tricerri and U Vanhecke (see [6], [7] and [11]).
The aim of this paper is to describe simply connected manifolds (M, g) admitting a homogeneous Riemannian structure T # 0, T 6 ~3, such that the related Ambrose-Singer connection has an algebraic curvature tensor R, i.e.
verifies the condition ~x,r,z R(X, Y)Z = 0.
Here, ~x,r,z denotes the cyclic sum over X, Y, Z and X, Y, Z belong to the Lie algebra 3/f(M) of the tangent vector fields on M. Observe that n = 3 is the minimum value of the dimension to have T # 0, and in this case (M, g) is not a direct product (see the definition at [8, p. 123]).
From [8] we already know the following results:
(A) n/> 4, M irreducible, R = 0 imply that M is a compact symmetric space with Betti number b3 # 0 (Theorem 1 in [8]). Unfortunately, in *This work was partially supported by MURST.