Riemannian manifolds with Einstein-like metrics (original) (raw)

Special metrics in G_2G_2G_2 geometry

2005

We review a recent series of G_2G_2G_2 manifolds constructed via solvable Lie groups obtained in math.DG/0409137. They carry two related distinguished metrics, one negative Einstein and the other in the conformal class of a Ricci-flat metric.

New Homogeneous Einstein Metrics of Negative Ricci Curvature

2000

We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of tendimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetric spaces of rank greater than one, preserving the (constant) Ricci curvature.

Remarks on Generalized Einstein Manifolds

2008

The usual Einstein metrics are those for which the first Ricci contraction of the covariant Riemann curvature tensor is proportional to the metric. Assuming the same type of restrictions but instead on the different contractions of the generalized covariant Gauss-Kronecker tensors Rp, leads to several generalizations of Einstein’s condition. In this paper, we treat some properties of these metrics. Mathematics Subject Classification (2000). 53C25, 58E11.

On Ricci curvature of metric structures on g-manifolds

2020

We study the properties of Ricci curvature of mathfrakg{\mathfrak{g}}mathfrakg-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the transverse manifold of the characteristic foliation are investigated. In particular, sufficient conditions are found under which the mathfrakg{\mathfrak{g}}mathfrakg-manifold can be a Ricci soliton or a gradient Ricci soliton. Finally, we obtain a amazing (non-existence) higher dimensional generalization of the Boyer-Galicki theorem on Einstein K-manifolds for a special class of abelian mathfrakg{\mathfrak{g}}mathfrakg-manifolds.

Some New Homogeneous Einstein Metrics on Symmetric Spaces

1996

We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank(M) > 1, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup G of Isom(M) acting transitively on M we nd all G-homogeneous (non-symmetric) Einstein metrics on M.

On semi-Riemannian manifolds satisfying some generalized Einstein metric conditions

arXiv (Cornell University), 2023

The difference tensor R • C − C • R of a semi-Riemannian manifold (M, g), dim M ≥ 4, formed by its Riemann-Christoffel curvature tensor R and the Weyl conformal curvature tensor C, under some assumptions, can be expressed as a linear combination of (0, 6)-Tachibana tensors Q(A, T), where A is a symmetric (0, 2)-tensor and T a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.

Homogeneous Einstein metrics on G_2/TG_2/TG_2/T

Proceedings of the American Mathematical Society, 2013

We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold M = G2/T. By computing a Gröbner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-Kähler invariant Einstein metrics. Thus G2/T turns out to be the first known example of an exceptional full flag manifold which admits at least one non-Kähler and not normal homogeneous Einstein metric.

Classification of Kantowski-Sachs Metric via Conformal Ricci Collineations

International Journal of Geometric Methods in Modern Physics, 2017

In this paper, we present a classification of the Kantowski–Sachs spacetime metric according to its conformal Ricci collineations (CRCs). Solving the CRC equations, it is shown that the Kantowski–Sachs metric admits 15-dimensional Lie algebra of CRCs when its Ricci tensor is non-degenerate and an infinite dimensional group of CRCs when the Ricci tensor is degenerate. Some examples of Kantowski–Sachs metric admitting nontrivial CRCs are presented and their physical interpretation is provided.