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Let H and S be the mean curvature and the squared length of the second fundamental form of submanifold M respectively. We prove that if M is an n(≥ 3)-dimensional complete submanifold with parallel mean curvature in Hn+p(−1), and if M (S−nH2)n/2dM < C(n,H), where H > 1 and C(n,H) is an explicit positive constant depending on n and H, then S ≡ nH2, i.e., M is the totally umbilical sphere Sn(1/ √ H2 − 1). Consequently, we show that if M is an n(≥ 3)-dimensional complete submanifold with parallel mean curvature in Sn+p, and if ∫ M (S−nH2)n/2dM < C ′(n), where C ′(n) is an explicit positive constant depending only on n, then M is the totally umbilical sphere Sn(1/ √ 1 + H2).

L<sup>n=2</sup> Pinching Theorem for Submanifolds with Parallel Mean Curvature in H<sup>n+p</sup>(-1)