## 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.

### MSC:

 53C20 Global Riemannian geometry, including pinching 57S25 Groups acting on specific manifolds
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### References:

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