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Journal of Mathematical Research Exposition May, 2008, Vol.28, No.2, pp.421–428 DOI:10.3770/j.issn:1000-341X.2008.02.024 Http:// On Totally Real Pseudo-Umbilical Submanifolds in a Complex Projective Space ZHANG Liang (College of Mathematics and Computer Science, Anhui Normal University, Anhui 241000, China) (E-mail: zhliang43@) Abstract Let Mn be a totally real submanifold in a complex projective space CP n+p . In this paper, we study the position of the parallel umbilical normal vector field of Mn in the normal bundle. By choosing a suitable frame field, we obtain a pinching theorem, in the case p 0, for the square of the length of the second fundamental form of a totally real pseudo-umbilical submanifold with parallel mean curvature vector. Keywords complex projective space; totally real submanifolds; pseudo-umbilical submanifolds; parallel mean curvature vector. Document code A MR(2000) Subject Classification 53C40 Chinese Library Classification O186.12 1. Introduction Let CP n+p be a 2(n + p )-dimensional complex projective space endowed with the Fubini- Study metric of constant holomorphic sectional curvature 4. Let M n be an n-dimensional sub- manifold in CP n+p . M n is called totally real if each tangent space of M n is mapped into the normal space by the complex structure J of CP n+p . In [1,2], in the case p = 0, a pinching theo- rem and the corresponding rigidity theorem for totally real minimal submanifolds were obtained. In [3], Du investigated the conditions under which totally real pseudo-umbilical submanifolds must be minimal for p = 0. In this paper, we study the totally real pseudo-umbilical subman