Non-negatively curved $C$-totally real submanifolds in a Sasakian manifold

Abstract

Several authors have investigated minimal totally real submanifolds in a complex space form and obtained many interesting results. Recently F. Urbano [6] and Y. Ohnita [4] have studied pinching problems on their curvatures and stated some theorems. On the other hand, in a (2n +1) -dimensional Sasakian space form of constant ^-sectional curvature c(> ―3), if a submanifold M is perpendicular to the structure vector field, then M is said to be C-totally real. For such a submanifold M, it is well-known that if the mean curvature vector field of M is parallel, then M is minimal. S. Yamaguchi, M. Kon and T. Ikawa [8] obtained that if the squared length of the second fundamental form of Mis less than n(n + Y)(c + 3)/4(2n ―1), then M is totally geodesic. Furthermore, D. E. Blair and K. Ogiue [2] proved that if the sectional curvature of M is a greater than (n―2) (c + 3)/4(2≪ ―1), then M is totally geodesic. In this paper, we consider a curvature-invariant C-totally real submanifold M in a Sasakian manifold with 37-parallelmean curvature vector field. Then M is not necessary minimal. Making use of methods of [3] and [4], we prove that if the sectional curvature of M is positive, then M is totally geodesic. In Sec. 1, we recall the differential operators on the unit sphere bundle of a Riemannian manifold. Sec. 2 is devoted to stating about fundamental formulas on a C-totally real submanifold in a Sasakian manifold. In Sec. 3, we prove Theorems and Corollaries. Throughout this paper all manifolds are always C°°, oriented, connected and complete. The author wishes to thank Professor S. Yamaguchi for his helo.