Abstract. This paper will prove that a bounded homogeneous domain is symmetric if and only if, in the Bergman metric, all sectional curvatures are nonpositive.Introduction. It is well known that a bounded symmetric domain has nonpositive sectional curvature in the Bergman metric. This paper is devoted to the converse, namely, that a bounded homogeneous domain (equivalently, homogeneous Siegel domain) which has nonpositive sectional curvature in the Bergman metric must be a symmetric domain. The proof uses the techniques of normal j-algebras [13], as well as results of Vinberg [16, 17], the first author [1][2][3], and, indirectly, Dorfmeister [5, 6]. The body of the paper is in two sections. In the first, we derive certain relations between the dimensions of the root spaces in an irreducible normal /algebra, which, in the presence of the additional curvature assumption, show the equality of the dimensions of all root spaces corresponding to certain roots for which no multiple is a root and the equality of the dimensions of all root spaces corresponding to roots which are half of another root. In the second section, we show that the first dimension result implies that the cone of the corresponding Siegel domain is self-dual while the two dimension results together imply that the domain is quasi-symmetric. The proof is then finished because [3] a quasi-symmetric domain with nonpositive sectional curvature in the Bergman metric is known to be symmetric. We remark that some of the subsidiary results mentioned above are not essentially new. However, for consistency of presentation, it is necessary to have them in the language of normal j-
