Riemannian manifolds with a Levi-Civita connection and constant Ricci curvature, or Einstein manifolds, were studied in the works of many mathematicians. This question has been most homogeneous case. In this direction, famous ones are results by D.V. Alekseevsky, M. Wang, V. Ziller, G. Jensen, H.Laure, Y.G. Nikonorov, E.D. Rodionov other At same time, studying little for case an arbitrary metri...

This paper is the continuation of [11] where the rotation construction of left-continuous triangular norms was presented. Here the class of triangular subnorms and a second construction, called rotation-annihilation, are introduced: Let T1 be a left-continuous triangular norm. If T1 has no zero divisors then let T2 be a left-continuous rotation invariant t-subnorm. If T1 has zero divisors then ...

We extend the results of Walters on the uniqueness of invariant measures with maximal entropy on compact groups to an arbitrary locally compact group. We show that the maximal entropy is attained at the left Haar measure and the measure of maximal entropy is unique.

In computational anatomy, organ’s shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie group is a manifold with a consistent group structure. Statistics on Riemannian manifolds have been well studied, but to use the statistical Riemann...

We discuss D-dimensional scalar field interacting with a scale invariant random metric which is either a Gaussian field or a square of a Gaussian field. The metric depends on d dimensional coordinates (where d < D). By a projection to a lower dimensional subspace we obtain a scale invariant non-Gaussian model of Euclidean quantum field theory in D − d or d dimensions.

It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the dimensional Heisenberg group. In this paper, we classify direct product of group Euclidean space dimension $n-3$ with $n \geq 4$, prove six such Lie automorphisms. Moreover show only one them is flat, other five are Ricci solitons but not Einstein. We also characterize flat ...

A natural generalization of analytic functions to n-dimensional Euclidean space are quasiregular mappings.(See [2] and [3].) An analogue of Theorem 1.1 for quasiregular mappings in John domains in Euclidean space appeared in [4]. Recently the analytical tools used in the proof of this result have been generalized to Carnot groups. We give an account of some of these advances and obtain an analo...

We obtain analytic solutions for the density contrast and the anisotropic pressure in a multi-dimensional FRW cosmology with collisionless, massless matter. These are compared with perturbations of a perfect fluid universe. To describe the metric perturbations we use manifest gauge invariant metric potentials. The matter perturbations are calculated by means of (automatically gauge invariant) f...

We investigate consequences of allowing the Hilbert space of a quantum system to have a time-dependent metric. For a given possibly nonstationary quantum system, we show that the requirement of having a unitary Schrödinger time-evolution identifies the metric with a positive-definite (Ermakov-Lewis) dynamical invariant of the system. Therefore the geometric phases are determined by the metric. ...

Conformal invariant new forms of p-brane and Dp-brane actions are proposed. These are quadratic in ∂X for the p-brane case and for Dp-branes in the Abelian field strength. The fields content of these actions are: an induced metric, gauge fields, an auxiliary metric and an auxiliary scalar field. The proposed actions are Weyl invariant in any dimension and the elimination of the auxiliary metric...