Abstract
Riemannian manifolds of sign-definite sectional curvature have been studied by many mathematicians due to the close relationship between the curvature and the topology of a Riemannian manifold.
We study Riemannian manifolds whose metric connection is a connection with vectorial torsion. The class of such connections contains the Levi-Civita connection. Although the curvature tensor of such a connection does not possess symmetries of the curvature tensor of the Levi-Civita connection, it is possible to define the sectional curvature. We investigate the question on relations between the sectional curvature of a connection with vectorial torsion and the sectional curvature of the Levi-Civita connection (Riemannian curvature). We also study the sign of sectional curvatures of connections with vectorial torsion. As an example, we consider Lie groups with left-invariant Riemannian metrics.
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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 6, pp. 86–92.
(submitted by S. K. Vodopyanov)
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Klepikov, P.N., Rodionov, E.D. & Khromova, O.P. Sectional Curvature of Connections with Vectorial Torsion. Russ Math. 64, 75–79 (2020). https://doi.org/10.3103/S1066369X20060110
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DOI: https://doi.org/10.3103/S1066369X20060110