Geometriae Dedicata 67: 233–243, 1997. 233 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. On Saddle Submanifolds of Riemannian Manifolds A. BORISENKO 1 , M. L. RABELO 2 and K. TENENBLAT 2 1 Geometry Department, Math.-Mech. Faculty, Kharkov State University, Pl. Svobodu 4, 310077-Kharkov, Ukraine. e-mail: borisenk@geom.kharkov.ua 2 Departamento de Matem´ atica, IE, Universidade de Bras´ ılia, 71910-900 Bras´ ılia, DF, Brasil e-mail: rabelo@mat.unb.br; keti@mat.unb.br (Received: 28 May 1996) Abstract. Saddle submanifolds are considered. A characterization of such submanifolds of Euclidean space is given in terms of sectional curvature. Extending results of T. Frankel, K. Kenmotsu and C. Xia, we determine under what conditions two complete saddle submanifolds of a complete connected Riemannian manifold , with nonnegative -Ricci curvature, must intersect. Moreover, if has positive -Ricci curvature and the dimension of a compact saddle submanifold satisfies a certain inequality then we show that the homomorphism of the fundamental groups 1 and 1 is surjective. Mathematics Subject Classifications (1991): 53A07, 53C21. Key words: saddle submanifolds, partial positive curvature. Hadamard [7] proved that on a complete surface with positive Gaussian curvature, every geodesic must meet every closed geodesic. In 1961, Frankel [4] generalized this theorem to Riemannian spaces of positive sectional curvature. He showed that two compact totally geodesic submanifolds 1 and 2 of an -dimensional Riemannian manifold of positive sectional curvature must necessarily intersect if the sum of their dimensions is at least equal to . As a consequence of this theorem, in 1966, Frankel [5] showed that if is complete and connected with strictly positive sectional curvature and is an -dimensional compact totally geodesic manifold immersed in , with 2 , then the homomorphism of the fundamental groups 1 1 is surjective. In 1981, Frankel’s first theorem was generalized to -saddle submanifolds [2] of a positively curved manifold. Recently Kenmotsu and Xia [8], [9] extended Frankel’s theorem to the case where the ambient Riemannian manifold has par- tially positive curvature. In this paper we generalize these last results to -saddle submanifolds. We first recall the definition of manifolds with partially positive curvature ([8], [9]). Let be an -dimensional Riemannian manifold. We say that has positive (resp. nonnegative ) -Ricci curvature at a point of if, for any 1 orthonormal tangent vectors 1 at we have 1 0 (resp. 0 , where denotes the sectional curvature of the plane spanned by and This work was partially supported by CNPq and FAPDF.