The Quarterly Journal of Mathematics Advance Access published online on November 2, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han029
ON SOME CONFORMAL MINIMAL 2-SPHERES IN A COMPLEX PROJECTIVE SPACE
Department of Mathematics, Graduate University, Chinese Academy of Sciences, Beijing 100049, China
Corresponding author. E-mail: xxj{at}gucas.ac.cn
Received 9 November 2007;
revised 29 August 2008
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Abstract |
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In this paper, the geometry of a linearly full conformal minimal 2-sphere S2 immersed in a complex projective space CPn which satisfies various conditions is studied. Let 
1(p) be the first normal space of S2 at the point p, and let Tp
S2 = 1(p) 
2(p) for p 
S2. We prove that S2 is of constant Kähler angle if and only if J1(p) 
Tp S2 for all p S2, where J is the complex structure of CPn. Furthermore, we prove that (i) S2 is totally geodesic in CP2 if J 1(p) Tp S2 for all p S2; (ii) S2 is either a holomorphic curve in CPn or the first element of the Veronese sequence, up to an isometry of CPn, if J1(p) 1(p) for all p S2; (iii) S2 is totally real if J1(p) 2(p) for all p S2. It is also proved that S2 is either an element of the Veronese sequence in CP2 or a totally real curve of constant curvature 1/3 in CP4 if its second fundamental form is parallel.
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