We study the static Einstein equation coupled with a scalar field with axial symmetry. The zero curvature equation both for the scalar field and a part of the metric in the scalar-Einstein equation enables one to obtain soliton solutions for both the scalar field and the metric. Of the infinite series of solutions for the metric and the scalar field we focus on the solution composed of 4-solito...

In this paper we describe the Einstein-Kähler metric for the Cartan-Hartogs domains which are the special case of the Hua domains. First of all, we reduce the Monge-Ampère equation for the metric to an ordinary differential equation in the auxiliary function X = X(z, w). This differential equation can be solved to give an implicit function in X. Secondly, for some cases, we obtained explicit fo...

In this paper we study the structure of standard Einstein solvmanifolds of arbitrary rank. Also the validity of a variational method for finding standard Einstein solvmanifolds is proved.

One of the most interesting questions in Riemannian geometry is that of deciding whether a manifold admits curvatures of certain kinds. More specifically, one might want to know whether some given manifold admits a canonical metric, i.e. one with constant curvature of some form (sectional curvature, scalar curvature, etc.). (This will in fact have many topological implications.). One such probl...

A generalized flag manifold is a homogeneous space of the form G/K, where K is the centralizer of a torus in a compact connected semisimple Lie group G. We classify all flag manifolds with four isotropy summands by the use of t-roots. We present new G-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics. 2000 Mat...

The only justification for the Einstein velocity addition law ‎appeared to be its empirical adequacy‎, ‎so that the ‎intrinsic beauty and harmony in Einstein addition remained for a long time ‎a mystery to be conquered‎. ‎Accordingly‎, ‎the aim of this expository article is to present ‎(i) the Einstein relativistic vector addition‎, ‎(ii) the resulting Einstein scalar multiplication‎, ‎(iii) th...

We show that Husain's reduction of the self-dual Einstein equations is equivalent to the Pleba´nski equation. The Bäcklund transformation between these equations is found. Contact symmetries of the Husain equation are derived.

We present a detailed derivation of the closed-form expression for the diffusion coefficient that was initially obtained by Einstein [4]. The present derivation does not make use of a fictitious force as did the original Einstein derivation, but instead concentrates directly on establishing a dynamic equilibrium between the forces of pressure and friction acting on a Brownian particle. This app...