On Sectional Curvature of Metric Connection with Vectorial Torsion
УДК 514.765
Abstract
Papers of many mathematicians are devoted to the study of semisymmetric connections or metric connections with vector torsion on Riemannian manifolds. This type of connectivity is one of the three main types discovered by E. Cartan and finds its application in modern physics, geometry, and topology of manifolds. Geodesic lines and the curvature tensor of a given connection were studied by I. Agricola, K. Yano, and other mathematicians. In particular, K. Yano proved an important theorem on the connection of conformal deformations and metric connections with vector torsion. Namely: a Riemannian manifold admits a metric connection with vector torsion and the curvature tensor being equal to zero if and only if it is conformally flat. Although the curvature tensor of a hemisymmetric connection has a smaller number of symmetries compared to the Levi-Civita connection, it is still possible to define the concept of sectional curvature in this case. The question naturally arises about the difference between the sectional curvature of a semisymmetric connection and the sectional curvature of a Levi-Civita connection.This paper is devoted to the study of this issue, and the authors find the necessary and sufficient conditions for the sectional curvature of the semisymmetric connection to coincide with the sectional curvature of the Levi-Civita connection. Non-trivial examples of hemisymmetric connections are constructed when possible.
Downloads
Metrics
References
Cartan E. Sur les varietes a connexion affine et la theorie de la relativite generalisee (deuxieme partie) // Ann. Ecole Norm. Sup. 1925. Vol. 42.
Schouten J.A. Ricci-Calculus.An intro-dustion to tensor analisis and geometrical Application Springer-Verlag. Berlin-Cottingen-Heidelberg, 1954.
Ivanov S., Parton M., Piccinni P. Loccaly conformal parallel G2- and Spin(7)-structures // Math. Res. Lett. 2006. Vol. 13.
Agricola I. The Srni lectures on non-integable geometries with torsion // Arch. Math. 2006. Vol. 42.
Галаев С.В. Почти контактные метрические пространства с N -связностью // Изв. Сарат. ун-та. 2015. Т. 15. Вып. 3.
Паньженский В.И, Климова Т.Р. Контактная метрическая связность на группе Гейзенберга // Изв. вузов. Матем. 2018. № 11.
Yano K. On semi-symmetric metric connection // Revue Roumame de Math. Pure et Appliquees. 1970. Vol. 15.
Agricola I., Kraus M. Manifolds with vectorial torsion // Differential Geometry and its Applications. 2016. Vol. 46.
Barua B., Ray A. Kr. Some properties of a semi-symmetric metric connection in a Riemannian manifold // Indian J. pure appl. Math. 1985. Vol. 16, No 7.
De U. C., De B. K. Some properties of a semi-symmetric metric connection on a Riemannian manifold // Istanbul Univ. Fen. Fak. Mat. Der. 1995. Vol. 54.
Manuraj D. Manifolds Admitting a semi-symmetric metric connection and a generalization of Shur’s theorem // Int. J. Contemp. Math. Scientes. 2018. Vol. 3, No 25.
Milnor J. Curvature of left invariant metric on Lie groups. // Advances in mathematics. 1976.
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
