Related papers: Bochner's technique in Einstein's non-symmetric ge…
The third del Pezzo surface admits a unique Kaehler-Einstein metric, which is not known in closed form. The manifold's toric structure reduces the Einstein equation to a single Monge-Ampere equation in two real dimensions. We numerically…
Connes' non-commutative geometry (NCG) is a generalization of Riemannian geometry that is particularly apt for expressing the standard model of particle physics coupled to Einstein gravity. In a previous paper, we suggested a reformulation…
Given an exceptional compact simple Lie group $G$ we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of $G$ over flag manifolds with a certain kind of isotropy…
For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G with compact, smooth orbit space, we show the following rigidity result: The nilradical N of G acts polarly, and the N-orbits can be…
In this work, we investigate the geometry and topology of compact Einstein-type manifolds with nonempty boundary. First, we prove a sharp boundary estimate, as consequence we obtain under certain hypotheses that the Hawking mass is bounded…
For any $G$-invariant metric on a compact homogeneous space $M=G/K$, we give a formula for the Lichnerowicz Laplacian restricted to the space of all $G$-invariant symmetric $2$-tensors in terms of the structural constants of $G/K$. As an…
We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of…
This paper develops a method for solving Einstein's equation numerically on multi-cube representations of manifolds with arbitrary spatial topologies. This method is designed to provide a set of flexible, easy to use computational…
We observe that, for a Bismut Einstein metric, the (2,0)-part of Bismut Ricci form is an eigenvector of the Chern Laplacian. With the help of this observation, we prove that a Bismut Einstein metric with non-zero Einstein constant is…
We construct the homogeneous Einstein equation for generalized flag manifolds $G/K$ of a compact simple Lie group $G$ whose isotropy representation decomposes into five inequivalent irreducible $\Ad(K)$-submodules. To this end we apply a…
Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The infinitesimal generators of these symmetries are…
A maximally reduced system of equations corresponding to the twisting type N Einstein metrics is given. When the cosmological constant $\lambda\to 0$ they reduce to the standard equations for the vacuum twisting type N's. All the metrics…
Using Cartan's equivalence method for point transformations we obtain from first principles the conformal geometry associated with third order ODEs and a special class of PDEs in two dimensions. We explicitly construct the null tetrads of a…
A Riemannian manifold $(M,g)$ is called \emph{weakly Einstein} if the tensor $R_{iabc}R_{j}^{~~abc}$ is a scalar multiple of the metric tensor $g_{ij}$. We give a complete classification of weakly Einstein hypersurfaces in the spaces of…
We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing…
In this paper, a Riemannian geometry of noncommutative super surfaces is developed which generalizes [4] to the super case. The notions of metric and connections on such noncommutative super surfaces are introduced and it is shown that the…
It is well-known that every 6-dimensional strictly nearly K\"{a}hler manifold $(M,g,J)$ is Einstein with positive scalar curvature $scal>0$. Moreover, one can show that the space $E$ of co-closed primitive (1,1)-forms on $M$ is stable under…
Various types of Lagrange and Finsler geometries and the Einstein gravity theory, and modifications, can be modelled by nonholonomic distributions on tangent bundles/ manifolds when the fundamental geometric objects are adapted to nonlinear…
Based on the view that Einstein's theory can be interpreted as a gauge theory of Lorentz group, we decompose the gravitational connection (the gauge potential of Lorentz group) $\vGm_\mu$ into the restricted connection made of the potential…
We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $\Omega^1$ depending on the Hamiltonian $p^2/2m + V(x)$, and a flat quantum connection $\nabla$ with torsion such that a…