Ricci curvature of manifolds of the Kaluza-Klein type
Let \(G\) be a one-dimensional connected Lie group, \(f : E\to M\) a principal \(G\)-bundle and \(g\) the \(G\)-invariant Riemannian metric on the total space \(E\). The pair \((E, g)\) is called a manifold of Kaluza-Klein type. The second author has studied the sectional curvature of these manifolds [Izv. Vyssh. Uchebn. Zaved., Mat. 1997, No. 9, 76-82 (1997; Zbl 0909.53034)]. In the present paper the following results for the Kaluza-Klein manifolds are obtained: The Ricci curvature \(\text{Ri}(X,X)\) of a tangent vector \(X\) may be expressed by a matrix \(C_{ij}\), defined in terms of Laplace \((C_{11})\), \(\text{div} (C_{12})\), Ri and He\((C_{22})\) operators, such that: Ri is positive iff \(C_{11}\) and \(\text{det }C\) are positive. If \((E, g)\) is a complete Riemannian manifold whose Ricci curvature is bounded below by a positive number, then the characteristic class of \(f\) is non-zero; the rang of the image of Hurewicz homomorphism is positive. If \((E, g)\) is a complete Riemannian manifold with positive Ricci curvature, \(\dim E=3\), then \(E\) and \(M\) are compact manifolds. But for \(\dim E=3\), there are complete Riemannian manifolds with positive Ricci curvature whose fundamental group is \(\mathbb{Z}\) [\textit{A. L. Besse}, Einstein manifolds, Springer, Berlin (1987; Zbl 0613.53001)]. If \((E, g)\) is a complete Riemannian manifold with negative Ricci curvature, then the base space \(M\) is non-compact.
scientific article; zbMATH DE number 1752650