Three-dimensional conformally flat pseudo-symmetric spaces of constant type. (English) Zbl 1054.53060

In a previous work, the reviewer and M. Sekizawa gave a complete explicit classification of so-called pseudo-symmetric spaces of constant type in dimension \(3\). This means that all corresponding metrics are given by finite formulas depending on arbitrary functions (except the so-called elliptic type where only special classes of examples are given in the explicit form). The spaces in question are characterized by the properties that one of the Ricci eigenvalues is constant and two others are arbitrary but equal. One works here with systems of nonlinear PDE of second order.
In the present paper the question, which of these spaces are conformally flat, is explicitly solved. It is shown that all these spaces are of “planar type” and they are locally warped products of a real line and a \(2\)-dimensional space form. Thus, they depend on one arbitrary function of one variable.


53C20 Global Riemannian geometry, including pinching
53C35 Differential geometry of symmetric spaces
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