Math. Ann. 267, 433 437 (1984) Am 9 Springer-Verlag 1984 Rigidity of Minimal Submanifolds in Space Forms J. L. M. Barbosa 1, M. Dajczer 2, and L. P. Jorge 1 1 IMPA, Est Dona Castorina 110, 22460 Rio de Janeiro, Brasil 2 Departamento de Matematica, Universidad Federal do Cear~, Fortaleza, Brasil 1. Introduction (1.1) Let c be a real number. Represent by A4"(c) a n-dimensional space form of curvature c. Let M N be a N-dimensional connected Riemannian manifold. The question that served as the starting point for this paper was to find simple conditions on the metric of M so that, if f:M"~ffl"+P(c), p>= 1, is an isometric minimal immersion then f is rigid in the following sense: given another minimal immersion g: M" ~ A4"+ q(c), q >= p, then there exists a rigid motion T of M~ + q(c) such that g = Tof, (~l"+P(c) being considered as a totally geodesic submanifold of In what follows we present a satisfactory answer to this question when n> 3 and p = 1 by imposing a restriction on the possible values of the nullity of the curvature tensor of M. (1.2) Let's recall that the nullity of the curvature tensor of M is a function p:M~Z that associates to each p E M the dimension of kerpR, the kernel of the curvature tensor R of M at p. That is, p(p): dim{X~ TrM;R(X , Y)=c(Y, 9 )X-c(X,. )Y for all Y~ TpM} where (., 9 ) stands for the metric of M. The result we obtain can be stated as follows: (1.3) Theorem. Let M"(n > 3) be a connected Riemannian manifold and p be a point of M. Assume #(p)<n-3. If f:M"~M"+l(c) and g:M"--*ffl"+k(c) are isometric minimal immersions then there is a rigid motion T of M"+k(c) such that g = To f, ( ~1"+ 1(c) being considered as a totally geodetic submanifold of 371" +R(c)). (l.4) The hypothesis of this theorem can not be weakened as shown by the following examples: Let f: M z ~R 3 and g:M"~ R" + k be isometric minimal immersions. Denote by J, the conjugate immersion of f As in [5] we can define fo: M2---,,R6= R3(~ R 3