Curvature Properties of Four-Dimensional Generalized Symmetric Spaces

Abstract

The curvature of four-dimensional generalized symmetric spaces is completely described. In particular, Einstein-like metrics on these spaces are classified. Interesting behaviours are found in the Lorentzian case and in one of the pseudo-Riemannian ones. Mathematics Subject Classification (2000). 53C50, 53C20, 53C30.

Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces

“…By reversing the sign of the metric if necessary, we may assume that the metric is Riemannian. The following result was proven in [2].…”

“…The possible signatures of ds 2 are (4, 0), (0, 4) and (2,2). The space is of order 3 and a typical symmetry of order 3 centered at the origin is…”

“…They are divided into four classes, named type A, B, C and D and the (pseudo-)Riemannian metrics can have any signature. In [2], the Riemann-Christoffel curvature tensors of these spaces are computed. We will use the results of these computations, with some needed corrections, to study the hypersurfaces of these spaces.…”

Four-Dimensional Pseudo-Riemannian Generalized Symmetric Spaces Which are Algebraic Ricci Solitons

“…In [3] and [6], the Levi-Civita connection, the curvature tensor and the Ricci tensor of these spaces are computed; proving that type C is symmetric, that is, the covariant derivative of its curvature tensor vanishes at each point. We will use the results of these computations to study which types of these spaces are algebraic Ricci solitons.…”

“…We now recall the following result on the curvature tensor and the Ricci tensor of four-dimensional generalized symmetric spaces of type A (see [3] Now, let D ∈ Der(g), where g is the Lie algebra in (3.1). Put Using the lemma above, we now prove the following.…”