On the geometry of sub-Riemannian manifolds equipped with a canonical quarter-symmetric connection
| Autor: | S. V. Galaev |
|---|---|
| Jazyk: | English<br />Russian |
| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Дифференциальная геометрия многообразий фигур, Vol 54, Iss 1, Pp 64-77 (2023) |
| Druh dokumentu: | article |
| ISSN: | 0321-4796 2782-3229 |
| DOI: | 10.5922/0321-4796-2023-54-1-7 |
| Popis: | 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. |
| Databáze: | Directory of Open Access Journals |
| Externí odkaz: |
|