On the geometrical properties of solvable Lie groups

What is it about?

‎Lie groups have been created by Marius Sophus Lie‎, ‎Fredrich Engel and Felix Klein between 1888 and 1893‎. ‎In 1925-1926 by the development of modern quantum theory these concepts found their way into physics and today are fundamental tools in many areas such as; Analysis‎, ‎differential geometry‎, ‎number theory‎, ‎differential equations‎, ‎atomic structures‎, ‎quantum theory and high energy physics‎. The building blocks for Lie groups are solvable Lie groups‎. ‎Solvable Lie groups equipped with a‎ left invariant Riemannian metric‎ ‎play an‎ ‎important and fundamental role in many branches of mathematics‎, ‎e.g.‎, ‎homogeneous Einstein manifolds of non-positive curvature‎, ‎Calabi-Yau equations on 4-manifolds‎, constructions of Riemannian manifolds‎ ‎with exceptional holonomy‎, ‎etc‎. ‎Thus it is worthwhile to study geometric properties of solvable Lie groups to improve our knowledge of these spaces‎. ‎In this paper a class of solvable Lie groups with an arbitrary odd dimension which are equipped with left-invariant Riemannian and Lorentzian metrics are considered‎. ‎These Lie groups were first introduced by Bozek and their homogeneous Riemannian geodesics were investigated by Calvaruso‎, ‎Kowalski and Marinosci‎. ‎Also some other geometrical properties‎, ‎such as left-invariant‎ Ricci solitons‎, ‎harmonicity of invariant vector fields‎, ‎invariant contact structures and homogeneous‎ structures on these homogeneous spaces with dimension five were investigated by authors previously‎. ‎Our aim in‎ ‎this paper is to extend these geometrical properties for an arbitrary odd dimension in both Riemannian and‎ Lorentzian cases‎. ‎By this study all of descriptions of the homogeneous Lorentzian and Riemannian‎ structures and their types on these spaces are obtained‎. ‎Also the energy of an arbitrary left-invariant vector field X on‎ these spaces is calculated and in the Lorentzian case it is proved that no left-invariant unit time-like vector fields on these‎ ‎spaces are critical points for the space-like energy‎. ‎Moreover‎, ‎the non-existence of invariant contact‎ structures and left-invariant Ricci solitons on these homogeneous spaces are also proved‎.

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