复空间形式中全实2-调和子流形.pdf

Journal of Mathematical Research Exposition Aug., 2008, Vol. 28, No. 3, pp. 727–732 DOI:10.3770/j.issn:1000-341X.2008.03.038 Http:// 2-Harmonic Submanifolds in a Complex Space Form 1 2 ZHU Ye Cheng , SONG Wei Dong (1. College of Mathematics and Physics, Anhui Industry University, Anhui 243002, China; 2. College of Mathematics and Computer Science, Anhui Normal University, Anhui 241000, China) (E-mail: zhuyecheng929@) Abstract In the present paper, the authors study totally real 2-harmonic submanifolds in a complex space form and obtain a Simons’ type integral inequality of compact submanifolds as well as some relevant conclusions. Keywords 2-harmonic; minimal; totally geodesic; complex space form. Document code A MR(2000) Subject Classification 53C42 Chinese Library Classification O186.12 1. Introduction Following the tentative ideas of Eell and Lemaire, Jiang[1,2] studied 2-harmonic map on Riemannian manifolds. After that, many new results concerning 2-harmonic submanifolds come out in succession. In 1999, Sun and Zhong[3] discussed real 2-harmonic hypersurface in a complex projective space. In this paper, we study totally real 2-harmonic submanifolds in a complex space form, generalize the conclusion in [3], and obtain a series of results. 2. Preliminaries Let CN n be a complex space form[4], of complex dimension n, with the Fubini-study metric of c constant holomorphic sectional curvature c, and J be the complex structure of CN n . Among all c submanifolds of CN n , there are two typical classes: one is the class of holomorphic submanifolds