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Riemannian 3-manifolds with \(c\)-conullity two. (English) Zbl 0879.53034

Let \((M,g)\) be a Riemannian manifold and \(R\) its Riemannian curvature tensor. For \(c\in \mathbb{R}\), define \(R_c\) by \(R_c (X,Y)Z= c\{g(Y,Z)X- g(X,Z)Y\}\). Then the \(c\)-nullity space at \(x\in M\) is the linear subspace \(T_{x,c} M= \{X\in T_xM |(R -R_c) (X,Y)Z=0\) for all \(Y,Z \in T_xM\}\). The integer \(\dim M- \dim T_{x,c}M\) is called the \(c\)-conullity of \((M, g)\) at \(x\).
In this paper, the authors study in detail three-dimensional Riemannian manifolds of \(c\)-conullity two with respect to some \(c\). This is equivalent to the assumption that two principal Ricci curvatures \(\rho_1\) and \(\rho_2\) are equal (and may depend on the point) and \(\rho_3=2c\). In their study they restrict to the case where the scalar curvature is constant along the geodesics tangent to the one-dimensional \(c\)-nullity spaces. The main purpose is to get explicit formulas for the metrics. This was already done for \(c=0\), i.e., for semi-symmetric spaces. (See E. Boeckx, O. Kowalski and L. Vanhecke, “Riemannian manifolds of conullity two” (World Scientific, River Edge, NJ) (1996)] for more details and references.) Here, one restricts to the hyperbolic case \((c<0)\) and the elliptic case \((c>0)\).
Following the same method as used by Kowalski for the case where the principal Ricci curvatures are global constants (and based on the explicit solution of a system of differential equations), the authors determine the complete explicit classification for the hyperbolic case and a quasi-explicit classification for the elliptic case. In the last case, they give an explicit example. In both cases they consider the local isometry classes and determine on what kind of functions they depend.
Note that type number two hypersurfaces in real space forms provide examples of Riemannian manifolds with \(c\)-conullity two. The main of the authors is to use the metrics for the construction of interesting hypersurfaces. For the flat space form \(\mathbb{R}^{n+1}\), these hypersurfaces were already discussed by Cartan in relation with the study of isometric deformations. New classes of three-dimensional examples have been discovered recently by V. Hájková for the case \(\rho_3 =0\) in her doctoral dissertation [Charles University, Prague (1995)].

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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