The Singer invariant of homogeneous spaces

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We investigate the problems on the relation between the homogeneity or the local homogeneity of a Riemannian manifold and the curvature tensor R and its covariant derivatives ∇R, ∇^2R,・・・, which are essential local invariants of a Riemannian manifold and obtain the following results.1. The Singer invariant : Given a locally homogeneous space M, we can define a non-negative integer κ_M from the data of its curvature tensor and covariant derivatives, which is called the Singer invariant of M. Our first problem is to compute the Singer invariant of various kinds of homogeneous spaces. We determined or estimated the Singer invariant for the following cases : (a) low-dimensional cases, in particular 4-dimensional homogeneous spaces, (b) homogeneous hypersurfaces in a unit sphere, (c) generalized Heisenberg groups with left invariant metric. Up to recently, at our knowledge, there were only a few homogeneous spaces whose Singer invariants are known and their Singer invariants are all at most 1. Recently C. Meusers proves, by giving explicit examples of solvmanifolds with high Singer invariant, that the Singer invariant of a locally homogeneous Riemannian manifold can become arbitrarily high. It is a remarkable result. We think that it will be an interesting problem to characterize his examples in the frame work of the Singer invariant.2. Curvature homogeneous spaces whose curvature tensors have large symmetries : Given a curvature tensor R, we denote by G_0 the identity component of the Lie group consisting of linear isometrics which preserve R invariantly. We study the following problems " Classify locally homogeneous spaces or curvature homogeneous spaces whose G_0 are large ". We obtained the results for the following cases : (a) G_0 = SO R x SO (n-r) (b) G_0 = S0 (n - 2) (c) G_0 acts transitively on a unit sphere.