The left-invariant contact metric structure on the Sol manifold

V.I. Pan’zhenskii∗ , A.O. Rastrepina∗∗

Penza State University, Penza, 440026 Russia

E-mail: ∗kaf-geom@yandex.ru, ∗∗n.rastrepina@mail.ru

Received October 30, 2019

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DOI: 10.26907/2541-7746.2020.1.77-90

For citation: Pan’zhenskii V.I., Rastrepina A.O. The left-invariant contact metric struc ture on the Sol manifold. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2020, vol. 162, no. 1, pp. 77–90. doi: 10.26907/2541-7746.2020.1.77-90. (In Russian)

Abstract

Among the known eight-dimensional Thurston geometries, there is a geometry of the Sol manifold – a Lie group consisting of real special matrices. For a left-invariant Riemannian metric on the Sol manifold, the left shift group is a maximal simple transitive group of isometry. In this paper, we found all left-invariant differential 1-forms and proved that on the oriented Sol manifold there is only one left-invariant differential 1-form, such that this form and the left- invariant Riemannian metric together define the contact metric structure on the Sol manifold. We identified all left-invariant contact metric connections and distinguished flat connections among them. A completely non-holonomic contact distribution along with the restriction of a Riemannian metric to this distribution define the contact metric structure on the Sol manifold, and an orthogonal projection of the Levi-Chivita connection is a truncated connection. We obtained geodesic parameter equations of the truncated connection, which are the sub-geodesic equations, using a non-holonomic field of frames adapted to the contact metric structure. We revealed that these geodesics are a part of the geodesics of the flat contact metric connection.

Keywords: Sol manifold, contact metric structure, contact metric connection, sub-Riemannian geodesics

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