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This article is cited in 5 scientific papers (total in 5 papers)
The left-invariant contact metric structure on the Sol manifold
V. I. Pan'zhenskii, A. O. Rastrepina Penza State University, Penza, 440026 Russia
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.
Received: 30.10.2019
Citation:
V. I. Pan'zhenskii, A. O. Rastrepina, “The left-invariant contact metric structure on the Sol manifold”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 162, no. 1, Kazan University, Kazan, 2020, 77–90
Linking options:
https://www.mathnet.ru/eng/uzku1546 https://www.mathnet.ru/eng/uzku/v162/i1/p77
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Abstract page: | 157 | Full-text PDF : | 46 | References: | 18 |
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