POMPEIU'S PROBLEM ON SPACES OF CONSTANT CURVATURE By CARLOS A. BERENSTEIN AND LAWRENCE ZALCMAN 1. The purpose of this paper is to formulate certain extensions of earlier results [13], [14], valid for euclidean space, to spaces of constant curvature. These include the solution of the Pompeiu problem for geodesic balls, analogues of Delsarte's two-radius theorem, and a generalized Pizzetti formula. The viewpoint adopted here is entirely classical, motivated largely by the concrete geometric appeal of the present situation and the explicit nature of the formulas obtained. Further extensions, to the setting of symmetric spaces of rank 1, are also possible; we intend to discuss such generalizations at length, and from a different point of view, in a subsequent publication. Accordingly, the treatment given here is abbreviated and the analysis rather sketchy. Research for this paper was supported by National Science Foundation Grant MPS 75-06977. Preparation of the paper was carried out while the authors were variously at the University of Maryland, Brandeis University, the Weizmann Institute of Science, the Technion, and the Forschungsinstitut fiir Mathematik of the ETH. The support, financial and otherwise, of all these institutions is gratefully acknowledged. This paper is dedicated, with admiration and affection, to our friend Max Schiffer. 2. Let S = S(n, k) be a complete, simply connected n-dimensional Rieman- nian manifold of constant curvature k. Then S is uniquely determined, up to isometric equivalence [8]. Of course, when k = 0 we may take S = R" with the usual euclidean metric ds2=dx~+dx~+...+dx2,. For k~0, various realiza- tions are possible. Thus, if k > 0 we may take for S the n-dimensional sphere S"(R) of radius R = 1/V~, centered at the origin in R "§ with the induced euclidean metric; equivalently, S may be realized (via the stereographic representa- tion of S"(R) on R") as R" with metric 4R'ldxl = (1) as 2 (R2+lx -- 12)2, where 113 JOURNAL D'ANALYSE MATHI~MATIOUE, Vol. 30 (1976)