Annals of Global Analysis and Geometry 17: 129–150, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. 129 Minimal Surfaces in a Sphere and the Ricci Condition ⋆ THEODOROS VLACHOS Department of Mathematics, University of Ioannina, Ioannina 45110, Greece (E-mail: tvlachos@cc.uoi.gr) Abstract. In this paper we deal with minimal surfaces in a sphere which are locally isometric to a minimal surface in S 3 . We prove that a minimal surface in a sphere is locally isometric to a minimal surface in S 3 if the curvature ellipse has constant and positive eccentricity. Moreover, we prove the following rigidity result: a compact minimal surface M in S m , m ≤ 6, cannot be locally isometric to a minimal surface in S 3 unless M already lies in S 3 or M is flat and lies in S 5 . Mathematics Subject Classifications (1991): 53A10 Key words: curvature ellipse, eccentricity, minimal surface, Ricci condition 0. Introduction Let Q m c denote the m-dimensional simply connected space form of constant cur- vature c and let M be a minimal surface in Q m c equipped with the induced metric ds 2 with Gaussian curvature K ≤ c. It is well known that when m = 3, M satisfies the Ricci condition, that is, the Riemannian metric d ˆ s 2 = √ c − Kds 2 is flat at points where K<c. Conversely, every 2-dimensional Riemannian manifold with Gaussian curvature K<c which satisfies the Ricci condition can be realized locally by a continuous one-parameter family of isometric minimal immersions f θ into Q 3 c ,0 ≤ θ<π . This result was discovered by Ricci and generalized by Lawson [11]. Therefore, all minimal surfaces in Q 3 c which are locally isometric to a given minimal surface in Q 3 c are obtained from the family f θ ,0 ≤ θ<π , the so called associated family of minimal immersions. Pinl [15] gave an example of a minimal surface in R 4 which does not satisfy the Ricci condition. Lawson [12] raised the problem to classify those minimal surfaces in Q m c which are locally isometric to a minimal surface in Q 3 c . Obviously, a minimal surface in Q m c is locally isometric to a minimal surface in Q 3 c if and only if the Ricci condition is satisfied. In the case when c = 0, Lawson [12] solved the problem completely. Using the generalized Gauss map he proved that minimal surfaces in R m which are locally ⋆ This work was written during the author’s stay at the Technische Universität Berlin as a research fellow of the Alexander von Humboldt Foundation.