On a Holonomy Flag of Non-holonomic Distributions
- 17 Downloads
We give definition of a holonomy flag in subRiemannian geometry, a generalization of the Riemannian holonomy algebra, and calculate it for the 3D subRiemannian Lie groups for different connections. We rewrite and give new interpretation for the Codazzi equations for the (2,3)-distributions on SU(2) and the Heisenberg group. We calculate holonomy flag for Tanaka-Webster, Tanno, and Wagner connections.
KeywordsHeisenberg group SubRiemannian 3D Lie group Codazzi equations Holonomy flag Tanaka-Webster connection Tanno connection Wagner connection
Mathematics Subject Classification (2010)53B15 58E25
The author is grateful to A.A. Agrachev, D.V. Alekseevsky, and Ya.V. Bazaikin for the stimulating conversations. The publication was supported by the Ministry of Education and Science of the Russian Federation (Project number 1.8126.2017/8.9).
- 2.Agrachev A, Barilari D, Rizzi L. Curvature: a variational approach.Google Scholar
- 3.Agrachev A, Barilari D, Rizzi L. Sub-Riemannian curvature in contact geometry.Google Scholar
- 5.Besse A. 1987. Einstein manifolds.Google Scholar
- 10.Wagner VV. 1939. Differential geometry of nonholonomic manifolds (Rusian). VIII Internat. competitionon searching N.I. Lobachevsky prize (1937). Report, Kazan, 195–262.Google Scholar
- 11.Berestovskii VN. Curvatures of homogeneous sub-Riemannian manifolds. European J Math. https://doi.org/10.1007/s40879-017-0171-3.