Classification of 4-dimensional homogeneous D’Atri spaces

Classification of 4-dimensional homogeneous weakly einstein manifolds

“…First, we calculate the conditions for (G, g) to be a weakly Einstein manifold. From [1] we know the expression for the curvature tensor. We denote by A ij the elementary skew-symmetric operators whose corresponding action is given by the formulas A ij (e l ) = δ il e j − δ jl e i .…”

Cyclic Lorentzian Lie groups

“…For example, this yields four standard forms of three-dimensional unimodular Lorentzian Lie groups [15], while just one form occurs in Riemannian settings [13]; • that every subspace of a vector space endowed with a positive definite inner product, inherits a positive inner product, while a subspace of a Lorentzian vector space inherits an inner product that can be either positive definite, Lorentzian, or even degenerate. In particular, this fact yields the differences in the classifications of three-dimensional non-unimodular Lorentzian [8] and Riemannian [13] Lie groups, and of left-invariant Lorentzian [7] and Riemannian metrics [2] on fourdimensional Lie groups.…”

“…(2) {c 1 = p 1 = p 2 = q 1 = q 2 = q 3 = 0}, so that g = h × R trivially, and, taking into account Proposition 4.1, h = sl (2).…”

On inaudible curvature properties of closed Riemannian manifolds

“…In the four dimensional case it is still open whether the type A property implies local homogeneity. Four-dimensional homogeneous spaces of type A have recently been classified [1], [3].…”

“…In dimension four, partial classifications were obtained by J.T. Cho, K. Sekigawa and L. Vanhecke [22], [23], [8]; the classification of 4-dimensional homogeneous D'Atri spaces has recently been completed by the first author and O. Kowalski [1], [2], [3]. See [17] for references about D'Atri spaces and related topics.…”