On the geometry of sub-Riemannian manifolds equipped with a canonical quarter-symmetric connection
- DOI
- 10.5922/0321-4796-2023-54-1-7
- Pages
- 64-77
Abstract
In this article, a sub-Riemannian manifold of contact type is understood as a Riemannian manifold equipped with a regular distribution of codimension-one and by a unit structure vector field orthogonal to this distribution. This vector field is called a structural. On a sub-Riemannian manifold of contact type, a quarter-symmetric connection is defined, which is associated with an endomorphism that preserves the distribution of the sub-Riemannian manifold. It is proved that if the connection under study is metric, then the endomorphism associated to it is uniquely defined. The structure of the associated endomorphism is found. In the case when the structure vector field is a field of infinitesimal isometries, the quarter-symmetric connection is called the canonical N-connection. An expression is found for the curvature tensor of the canonical N-connection in terms of the Riemann curvature tensor. The properties of the Schouten curvature tensor are investigated, which provide, in particular, the necessary symmetries of the curvature tensor of an N-connection for its sectional curvature to be well-defined. A relation between the sectional curvature of the canonical N-connection and the sectional curvature of the Levi-Civita connection is found. Necessary and sufficient conditions are found under which the sectional curvature of the N-connection and the sectional curvature of the Levi-Civita connection coincide.
Reference
1. Bukusheva, A. V.: On geometry of Kenmotsu manifolds with N-connection. DGMF, 50, 48—60 (2019).
2. Bukusheva, A. V.: Non-holonomic Kenmotsu manifolds equipped with generalized Tanaka — Webster connection. DGMF, 52, 42—51 (2021).
3. Galaev, S. V.: Extended structures on codistributions of contact metric manifolds. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform. 16:3, 263—272 (2016).
4. Galaev, S. V.: Almost contact metric spaces with N-connection. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 15:3, 258—263 (2015).
5. Galaev, S. V.: Generalized Wagner’s curvature tensor of almost contact metric spaces. Chebyshevskii sb., 17:3, 53—63 (2016).
6. Galaev, S. V., Gokhman, A. V.: Almost symplectic connections on a nonholonomic manifold. Mathematics. Mechanics, 3, 28—31 (2001).
7. Klepikov, P. N., Rodionov, E. D., Khromova, O. P.: Sectional curvature of connections with vectorial torsion. Russ Math., 64, 75—79 (2020).
8. Agricola, I., Kraus, M.: Manifolds with vectorial torsion. Diff. Geom. and its App., 46, 130—146 (2016).
9. Barua, B., Ray, A. Kr.: Some properties of a semi-symmetric metric connection in a Riemannian manifold. Indian J. Pure App. Math., 16:7, 736—740 (1985).
10. Biswas, S. C., De, U. C.: Quarter-symmetric metric connection in an SP-Sasakian manifold. Commun. Fac. Sci. Univ. Ank. Ser. А1, 46, 49—56 (1997).
11. Cartan, E.: Sur les varieties a connexion affine et la theorie de la relative generalisee. Part II. Ann. Ec. Norm., 42, 17—88 (1925).
12. Galaev, S. V.: Intrinsic geometry of almost contact Kahlerian manifolds. Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 31:1, 35—46 (2015).
13. Golab, S.: On semi-symmetric and quarter-symmetric linear connections. Tensor (N. S.), 29, 249—254 (1975).
14. Yano, K.: On semi-symmetric metric connection. Rev. Roum. Math. Pure Appl., 15, 1579—1586 (1970).
15. Yano, K. Imai, T.: Quarter-symmetric metric connections and their curvature tensors. Tensor (N. S.), 38, 13—18 (1982).