Abstract
We prove that, among all Riemannian spaces of constant curvature, only three-dimensional spaces have torsion which is invariant under the group of motions. The torsion tensor in these spaces is covariantly constant and determines the torsion form. The ratio of the integral of this form over a bounded domain to its volume is a constant determining the torsion of the space. We introduce the notions of volume torsion and scalar torsion.
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K. Jano and S. Bochner, Curvature and Betti Numbers (Princeton, 1953; Inostr. Lit., Moscow, 1957).
L. P. Eisenhart, Continuous Groups of Transformations (Princeton, 1933; Inostr. Lit., Moscow, 1947).
S. Sternberg, Lectures on Differential Geometry (Englewood Cliffs, 1964; Mir, Moscow, 1970).
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Original Russian Text © V. I. Pan’zhenskii, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 5, pp. 754–757.
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Pan’zhenskii, V.I. Maximally movable Riemannian spaces with torsion. Math Notes 85, 720–723 (2009). https://doi.org/10.1134/S0001434609050125
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DOI: https://doi.org/10.1134/S0001434609050125