On the curvature of homogeneous Kähler metrics of bounded domains (Q913278)

An n-dimensional Riemannian manifold is called a harmonic space if in a neighborhood of each point \(m_ 0\) in M, there exists a harmonic function f(m) depending only on the distance of m from \(m_ 0\) [\textit{H. S. Ruse}, \textit{A. G. Walker} and \textit{T. J. Willmore}, Harmonic spaces (1961; Zbl 0134.392)] and [\textit{A. L. Besse}, Manifolds all of whose geodesics are closed (1978; Zbl 0387.53010)]. It was shown by \textit{E. T. Copson} and \textit{H. S. Ruse} in [Proc. R. Soc. Edinb., Sect. A 60, 117- 133 (1940)], by \textit{A. Lichnerowicz} in [Bull. Soc. Math. Fr. 72, 146- 168 (1944; Zbl 0060.385)] and by \textit{A. J. Ledger} in [Ph. D. Thesis, Univ. Durham (1954)] that the curvature tensor of a harmonic space must satisfy an infinite sequence of conditions \(H_ k\). In the present paper, the author investigates the conditions \(H_ 1\) and \(H_ 2\) on the Riemannian manifolds obtained from a bounded domain D in \(C^ n\), endowed with a homogeneous Kähler metric.