A method, due tó Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is s...

A review of the geometry of 3-dimensional contact metric manifolds shows that generalized Sasakian manifolds and η-Einstein manifolds are deeply interrelated. For example, it is known that a 3-dimensional Sasakian manifold is η-Einstein. In this paper, we discuss the relationships between several special classes of 3-dimensional contact metric manifolds which are generalizations of 3-dimensiona...

A method, due tó Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is s...

We summarize recent results on the construction of Killing forms on SasakiEinstein manifolds. The complete set of special Killing forms of the Sasaki-Einstein spaces are presented. It is pointed out the existence of two additional Killing forms associated with the complex holomorphic volume form of Calabi-Yau cone manifold. In the case of toric Sasaki-Einstein manifolds the Killing forms are ex...

We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the GaussBonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral...

We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss–Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integra...

We discuss the space of Einstein metrics, up to diffeomorphism, on a compact manifold. In particular we mention some heuristics on its dimension and some theorems on its compactness.

A para-Kähler manifold can be defined as a pseudoRiemannian manifold (M, g) with a parallel skew-symmetric paracomplex structures K, i.e. a parallel field of skew-symmetric endomorphisms with K = Id or, equivalently, as a symplectic manifold (M, ω) with a bi-Lagrangian structure L, i.e. two complementary integrable Lagrangian distributions. A homogeneous manifold M = G/H of a semisimple Lie gro...

1 We describe in this paper all simply connected Spin c manifolds carrying parallel and real Killing spinors. In particular we show that every Sasakian manifold (not necessarily Einstein) carries a canonical Spin c structure with Killing spinors.

The object of the upcoming article is to characterize paracontact metric manifolds conceding $m$-quasi Einstein metric. First we establish that if $g$ in a $K$-paracontact manifold metric, then constant scalar curvature. Furthermore, classify $(k,\mu)$-paracontact whose Finally, construct non-trivial example such manifold.