Automorphisms of Riemann-Cartan manifolds. (English. Russian original) Zbl 1337.53045

Math. Notes 98, No. 4, 613-623 (2015); translation from Mat. Zametki 98, No. 4, 544-556 (2015).
Summary: It is proved that the maximal dimension of the Lie group of automorphisms of an \(n\)-dimensional Riemann-Cartan manifold (space) \((M^n, g, \tilde{\nabla})\) equals \(n(n-1)/2+1\) for \(n > 4\) and, if the connection \(\tilde{\nabla}\) is semisymmetric, for \(n\geq 2\). If \(n = 3\), then the maximal dimension of the group equals 6. Three-dimensional Riemann-Cartan spaces \((M^3, g,\tilde{\nabla})\) with automorphism group of maximal dimension are studied: the torsion \(s\) and the curvature \(\tilde{k}\) are introduced, and it is proved that \(s\) and \(\tilde{k}\) are characteristic constants of the space and \(\tilde{k} = k - s^2\), where \(k\) is the sectional curvature of the Riemannian space \((M^3,g)\); a geometric interpretation of torsion is given. For Riemann-Cartan spaces with antisymmetric connection, the notion of torsion at a given point in a given three-dimensional direction is introduced.


53C20 Global Riemannian geometry, including pinching
57S25 Groups acting on specific manifolds
Full Text: DOI


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