Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds

We completely classify three-dimensional homogeneous Lorentzian manifolds, equipped with Einstein-like metrics. Similarly to the Riemannian case (E. Abbena et al., Simon Stevin Quart J Pure Appl Math 66:173–182, 1992), if (M, g) is a three-dimensional homogeneous Lorentzian manifold, the Ricci tensor of (M, g) being cyclic-parallel (respectively, a Codazzi tensor) is related to natural reductivity (respectively, symmetry) of (M, g). However, some exceptional examples arise. The author is supported by funds of MURST, GNSAGA and the University of Lecce.     

Homogeneous Einstein-Randers metrics on spheres

In this paper, we study the homogeneous Einstein-Randers metrics on spheres. It turns out that we can find out all the homogeneous non-Riemannian Einstein-Randers metrics on spheres. Furthermore, we obtain a complete classification of such metrics under isometries. Using this, we present a large number of homogeneous Einstein-Randers metrics of non-constant flag curvature.     

Finsler metrics on symmetric cones

Let V be a simple Euclidean Jordan algebra with an associative inner product and let be the corresponding symmetric cone. Let be the compact symmetric space of all primitive idempotents of V. We show that the function s(a,b) defined by is a (the automorphism group of )-invariant complete metric on and it coincides with a natural Finsler distance on We also show that the metric s(a,b) (strictly) contracts any (strict) conformal compression of . Received: 24 May 1999 / in final form: 15 March 1999     

Homogeneous Einstein metrics on an exceptional Berger space

Barnaul. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 1, pp 208–211, January–February, 1992.     

Homogeneous manifolds admitting non-Riemannian Einstein-Randers metrics

In this paper, we find some new homogeneous manifolds G/H admitting non-Riemannian EinsteinRanders metrics when G is the compact simple Lie group E6, or E7 or E8. In the beginning, we prove that these homogeneous manifolds admit Riemannian Einstein metrics. Based on these metrics, we obtain non-Riemannian Einstein Randers metrics on them.     

Killing forms on symmetric spaces

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that a compact simply connected symmetric space carries a non-parallel Killing p-form (p?2) if and only if it isometric to a Riemannian product Sk×N, where Sk is a round sphere and k>p.     

Visible actions on symmetric spaces

A visible action on a complex manifold is a holomorphic action that admits a J-transversal totally real submanifold S. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism σ such that σ|_S = id. In this paper we prove that for any Hermitian symmetric space D = G/K the action of any symmetric subgroup H is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold S. Our geometric results provide a uniform proof of various multiplicity-free theorems of irreducible highest weight modules when restricted to reductive symmetric pairs, for both classical and exceptional cases, for both finite- and infinite-dimensional cases, and for both discrete and continuous spectra.     

Parabolic symmetric spaces

We study here systems of symmetries on |1|-graded parabolic geometries. We are interested in smooth systems of symmetries, and we discuss non-flat homogeneous |1|-graded geometries. We show the existence of an invariant admissible affine connection under quite weak condition on the system.     

Quantum symmetric spaces

Let G be a semisimple Lie group, g its Lie algebra. For any symmetric space M over G we construct a new (deformed) multiplication in the space A of smooth functions on M. This multiplication is invariant under the action of the Drinfeld-Jimbo quantum group U_hg and is commutative with respect to an involutive operator . Such a multiplication is unique. Let M be a kählerian symmetric space with the canonical Poisson structure. Then we construct a U_hg-invariant multiplication in A which depends on two parameters and is a quantization of that structure.