Toward a Theory of Spaces of Constant Curvature

We construct some examples of three-dimensional metrics of constant curvature defined by the solutions of nonlinear integrable differential equations and their generalizations and study the properties of Riemann extensions of the metrics of constant curvature. We discuss the connection with the theory of normal Riemannian spaces. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 42–54, January, 2006.