Totally umbilical submanifolds of complex space forms have been classified by B.Y. Chen&K. Ogiue, , cf. Theorem 1, p. 225. Their classification relies on the earlier observation (cf. , Prop. 3.1, p. 260) that curvature-invariant submanifolds of a complex space form are either holomorphic or totally real. In  one extends these ideas to submanifolds of Sasakian space forms and obtains the following:
Theorem 17.1 Let M2m+1 be an odd-dimensional totally umbilical sub-manifold of a Sasakian space form M2m+1(c), 1 < m < n. If M2m+1 is tangent to the contact vector ξ of M2n+1(c) then M2m+1 is a Sasakian space form immersed in M2n+1(c) as a totally geodesic submanifold.
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