, who founded and headed the geometry department of the University of Kazan in 1937, greatly appreciated the scientific work of Elie Cartan, the eminent French mathematician. (The biography of Shirokov is given in [24].)In 1937, the Kazan physics and mathematics society organized the 8th competition for the prize of N. I. Lobachevsky, and E. Cartan became a laureate of this competition.Shirokov arranged the translation of a number of Cartan's works. Some of these translations were pubfished in 1939, and others came off the press in 1962 ([1, 7]). Cartan's book The Theory of Spinor~ [8], translatecl by Shirokov, was published in 1947.In his scientific work, Shirokov payed special attention to symmetric spaces, the study of which he independently started in 1925. Shortly afterwards, the theory of these spaces was deeply developed by Cartan in connection with the theory of Lie groups (see [9]).In [23] (see also [24]), Shirokov posed the problem of classifying types of spaces with a positive-definite metric and a covariantly constant curvature tensor. He showed theft in the 3-dimensional case these spaces were either spaces of constant curvature or reducible spaces with a linear element, which takes the following form in suitable locM coordinates:He also posed the problem of studying conformally Euclidean symmetric spaces, which was completely solved in [25]. Abandoning the requirement that the metric be positive definite, he obtained the following results.A conformally Euclidean reducible space can be decomposed only into two subspaces. If each of them has dimension exceeding 1, then both of them are of constant curvature. Moreover, their scalar curvatures are equal in absolute value and opposite in sign. If one of the subspaces is one-dimensional, then the other is of constant curvature.In Pdemannian spaces with positive-definite linear elements, the above-ment_;oned spaces, together with the spaces of constant curvature, exhaust all conformally Euclidean symmetric spaces.For Riemanniaaa spaces with indefinite linear elements there exist other types of conformally Euclidean symmetric spaces.In fact, the curvature tensor of a conformaUy Euclidean space has the form and the requirement that this tensor be covariantly constant leads to the conditionIf we then rewrite the metric tensor of the required space in tee canonical form gij --hi j,
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