We study curvature homogeneous spaces or locally homogeneous spaces whose curvature tensors are invariant by the action of “large" Lie subalgebras of . In this paper we deal with the cases of
We show that the Singer invariant of a four-dimensional homogeneous space is at most .
A six-parameter family is constructed of (algebraic) Riemannian curvature tensors in dimension four which do not belong to any curvature homogeneous space. Also a general method is given for a possible extension of this result.
Using the Plücker map between grassmannians, we study basic aspects of classic grassmannian geometries. For ‘hyperbolic’ grassmannian geometries, we prove some facts (for instance, that the Plücker map is a minimal isometric embedding) that were previously known in the ‘elliptic’ case.
A compact K¨ahlerian manifoldM of dimension n satisfies hp,q(M) = hq,p(M) for each p, q.However, a compact complex manifold does not satisfy the equations in general. In this paper, we consider duality of Hodge numbers of compact complex nilmanifolds.
We prove that there is exactly one homothety class of invariant Einstein metrics in each space defined below.