C-TOTALLY REAL SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE IN λ-SASAKIAN SPACE FORMS

Abstract. In this paper, we prove a generalized integral inequality for an n-dimensional oriented closed C-totally real submanifold M with parallel mean curvature vector h in a (2m + 1)-dimensional closed λ-Sasakian space form M̃(c) of constant ϕ-sectional curvature c with 0 < c ≤ λ, n ≥ 2 and if a tensor φ related to h and the second fundamental form satisfies a certain inequality. As a consequence we obtain that M is totally umbilic or minimal with S = (n(c + 3λ) + (c − λ))/6, which generalize the Theorem 3 of [10]. Finally, we prove that if M is f-pseudo-parallel in a (2n + 1)-dimensional λ-Sasakian space form with f ≥ (n(c + 3λ) + (c − λ))/4n, then M is totally geodesic, which generalize the Theorem 1 of [13], when λ = 1. 1