Using the expression for the Laplacian of the square of the length of the second fundamental form of conformally fiat minimal C-totally real submanifolds of a Sasakian space form pinchings for scalar curvature and sectional curvature are obtained which imply that the submanifolds must be totally geodesic.
1. STATEMENT OF RESULTS
Let )~r 2"+ 1(~) be a (2n + 1)-dimensional Sasakian space form of constant q~-sectional curvature ~ and let M" be an n-dimensional minimal C-totally real submanifold of ~2~+ 1(~).
The following theorem is well-known (see, e.g., ).
THEOREM A. M ~ is totally geodesic if and only if one of the following holds: (a) K = ΒΌ(~ + 3), (b) p = ΒΌn(n -1)(~ + 3), where K and p are respectively the sectional and the scalar curvature of M. S. Yamaguchi, M. Kon and T. Ikawa proved the following [8]. THEOREM B. If M ~ is compact, then n2(n--2) ,~ . P> ~n2 1)~c-t" 3) implies that M is totally 9eodesic. THEOREM C. If M has constant scalar curvature n2(n -2),~ p> z tc*3), then M is totally 9eodesie. THEOREM D. If M has constant sectional curvature c, then either c=ΒΌ(~+3) or c <<, O.
Generalizations for a higher codimension and for submanifolds with a parallel mean curvature vector were obtained by Ludden et al. [6] and by K. Yano and M. Kon [9]. D. E. Blair and K. Ogiue proved the following [3].