Homogeneous geodesics in pseudo-Riemannian nilmanifolds

Abstract

Abstract. We study the geodesic orbit property for nilpotent Lie groups N when endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When N acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric is bi-invariant. Assuming N is 2-step nilpotent and with non-degenerate center we give algebraic conditions on the Lie algebra n of N in order to verify that every geodesic is the orbit of a one-parameter subgroup of N ⋊ Auto(N ). In addition we present an example of an almost g.o. space such that for null homogeneous geodesics, the natural parameter of the orbit is not always the affine parameter of the geodesic.

Geodesic orbit Riemannian structures on Rn

“…Various constructions of geodesics that are orbits of one-parameter isometry groups, and interesting properties of geodesic orbit metrics can be found in [12,15,22,26,32] and in the references therein. For generalizations to pseudo-Riemannian manifolds, see [5,13,14] and the references therein; for generalizations to Finsler manifolds, see [31].…”

Geodesic orbit metrics on homogeneous spaces constructed by strongly isotropy irreducible spaces

“…For left-invariant geodesic orbit metrics on Lie groups, one can refer to [9,10] and [27]. For the generalizations to pseudo-Riemannian manifolds and Finsler manifolds, see [7,12,13,22,38] and the reference therein.…”

Automorphism groups of pseudo $H$-type algebras

“…These algebras are some special examples of metric Lie algebras, studied in [4,15,19,20,22]. The pseudo H-type Lie groups is a fruitful source for study of geometry with non-holonomic constrains or nilmanifolds [18,21,34], symmetric spaces and harmonic spaces [7,8,14,39], differential operators on Lie groups [6,9,38,40].…”