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Pseudo-symmetric spaces of constant type in dimension three – elliptic spaces. (English) Zbl 0889.53026
Let \((M,g)\) be a Riemannian manifold and \(R\) its Riemannian curvature tensor. Then \((M,g)\) is called a pseudo-symmetric space of constant type \(c\) if for arbitrary vector fields \(X,Y\) there holds \(R(X,Y) \cdot R= c((X\wedge Y) \cdot R)\) where \(X\wedge Y\) denotes the endomorphism of the tangent bundle defined by \((X\wedge Y) Z=g(Y,Z) X-g(X,Z)Y\). The dot denotes derivation on the tensor algebra. For three-dimensional \((M,g)\) this condition is equivalent to \(\rho_1= \rho_2\), \(\rho_3= 2c\) where the \(\rho_i\), \(i=1,2,3\), are the eigenvalues of the Ricci operator. The condition \(\rho_1= \rho_2= \rho_3\) means that \((M,g)\) is of constant curvature. Furthermore, \(\rho_3 =0\) means that \((M,g)\) is semi-symmetric. For arbitrary dimension, the class of semi-symmetric spaces has been studied thoroughly be several researchers (see [E. Boeckx, O. Kowalski and L. Vanhecke, ‘Riemannian manifolds of conullity two’ (World Scientific, Singapore) (1996)] for a survey and further references).
The case of three-dimensional semi-symmetric spaces has been studied in detail by the first author who determined explicit forms for the metrics in almost full generality and who also considered in detail the local isometry classes of metrics on these spaces (see also the above-mentioned book). This method is based on an explicit solution of a particular system of partial differential equations. Using a similar method, the first author also treated the case where \(\rho_1 = \rho_2\) and \(\rho_3\) are constants.
In a former paper, the authors of the paper under review started their study of the three-dimensional pseudo-symmetric spaces of constant type, in particular for the special “non-elliptic” case (see also the above-mentioned book for further details). In this cae they gave a complete solution for the system of partial differential equations. Now, they continue their study and consider the more complicated “elliptic” case for which they give a quasi-explicit solution. Moreover, they prove that in the generic case the local isometry classes of metrics depend essentially on three arbitrary functions of two variables. They illustrate the theory by means of a construction of an explicit family of examples.

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)