LEFT-INVARIANT PSEUDO-EINSTEIN METRICS ON LIE GROUPS

In this article, we focus on left-invariant pseudo-Einstein metrics on Lie groups. To begin with, we give some examples of pseudo-Einstein metrics on Lie groups. Also we calculate the Levi-civita connection, and then Ricci tensor associated with left-invariant pseudo-Riemannian metrics on the unimodular Lie groups of dimension three. Furthermore, we show that the left-invariant pseudo-Einstein metric on SL(2) is unique up to a constant. At last, we study the left-invariant pseudo-Riemannian metrics on compact Lie groups and classify the pseudo-Einstein metrics on the low-dimensional compact Lie groups.     

Hyper-Kähler Metrics Conformal to Left Invariant Metrics on Four-Dimensional Lie Groups

Let g be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold (M,). We show that when the isometry group I(M,g) contains a subgroup G acting simply transitively on M by hypercomplex isometries, then the metric g is conformal to a hyper-Kähler metric. We describe explicitely the corresponding hyper-Kähler metrics, which are of cohomegeneity one with respect to a 3-dimensional normal subgroup of G. It follows that, in four dimensions, these are the only hyper-Kähler metrics containing a homogeneous metric in its conformal class.     

Slowly rotating white holes

A model of gravitationally anticollapsing objects, white holes, is constructed on the basis of the Kerr metric in the limit of small rotation with a corresponding interior metric. The extended space-time manifold is considered and the spectral shift of radiation from the point of view of a remote observer is calculated for different parameters of such white holes.     

On the Existence of a Gyroscope in Spaces with Affine Connections and Metrics

Conditions for the existence of a gyroscope in spaces with affine connections and metrics are found. They appear as special types of Fermi-Walker transports for vector fields, lying in a subspace, orthogonal to the velocity vector field (a non-null contravariant vector field) of an observer.     

Degenerate Hermitian Manifolds

The geometry of almost complex manifolds with degenerate indefinite Hermitian metrics is studied.     

Scalar Flat Metrics of Eguchi-Hanson Type

We use a new method to construct a class of asymptotically locally flat, scalar flat metrics. These metrics were constructed via algebraic geometry method by LeBrun before and provide counterexamples to the generalized positive action conjecture of Hawking and Pope.     

Definite signature conformal holonomy: A complete classification

This paper aims to classify the holonomy of the conformal Tractor connection, and relate these holonomies to the geometry of the underlying manifold. The conformally Einstein case is dealt with through the construction of metric cones, whose Riemannian holonomy is the same as the Tractor holonomy of the underlying manifold. Direct calculations in the Ricci-flat case and an important decomposition theorem complete the classification for definitive signature.     

Scalar curvature and projective compactness

Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) whereas its projective structure does, then the metric is projectively compact of order 2; this order is a measure of volume growth towards infinity. This implies a host of results including that the metric satisfies asymptotic Einstein conditions, and induces a canonical conformal structure on the boundary. Underpinning this work is a new interpretation of scalar curvature in terms of projective geometry. This enables us to show that if the projective structure of a metric extends to the boundary then its scalar curvature also naturally and smoothly extends.     

Calculation of Curvature Vacuum Correlations in R-Gravity

Under the flat Minkowski space-time background, using the perturbative expansion of the metric density, we calculate the expressions of the leading terms of several two-point curvature vacuum correlation functions in N-dimensional R-gravity, resulting in that the contributions of the leading terms of the curvature vacuum correlation functions are all vanishing.