In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the nilradical is of two-step. New examples of Einstein solvmanifolds with three-st...

The notion of Gelfand pair (G, K) can be generalized by considering homogeneous vector bundles over G/K instead the space and matrix-valued functions scalar-valued functions. This gives definition commutative bundles. Being a is necessary condition for being bundle. In case when nilmanifold having square-integrable representations, big family was determined in [Transform. Groups 24 (2019), no. ...

In this paper we first determine minimal models of nilmanifolds associated with given rational nilpotent Lie algebras. Then we study some properties of nilmanifolds through their associated Lie algebras and minimal models. In particular, we will see that a minimal model of a nilmanifold is formal if and only if it is a torus, and thus a non-toral nilmanifold has no complex structure which is bi...

Abstract We study the quaternionic Calabi–Yau problem in HyperKähler manifolds with torsion geometry, introduced by Alesker and Verbitsky [ 5], on eight-dimensional two-step nilmanifolds $M$ an Abelian hypercomplex structure. show that these Monge–Ampère equation can always be solved for any data are invariant under action of a three-torus.

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity has important classes of manifolds as special cases. Those include weakly symmetric naturally reductive manifolds. corresponding results for indefinite metric are much more delicate than signature, but the last few years structural were proved Lorentz Here, we carry out a major step analysis nil...