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Lagrangian submanifolds satisfying a basic equality

Published online by Cambridge University Press:  24 October 2008

Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027 U.S.A. E-mail address: bychen@math.msu.edu
Luc Vrancken
Affiliation:
Departement Wiskunde, Celestijnenlaan 200 B, B-3001 Leuven, Belgium e-mail address: luc.vrancken©wis.kuleuven.ac.be
Corresponding

Abstract

In [3], B. Y. Chen proved that, for any Lagrangian submanifold M in a complex space-form Mn(4c) (c = ± 1), the squared mean curvature and the scalar curvature of M satisfy the following inequality:

He then introduced three families of Riemannian n-manifolds and two exceptional n-spaces Fn, Ln and proved the existence of a Lagrangian isometric immersion pa from into ℂPn(4) and the existence of Lagrangian isometric immersions f, l, ca, da from Fn, Ln, , into ℂHn(− 4), respectively, which satisfy the equality case of the inequality. He also proved that, beside the totally geodesie ones, these are the only Lagrangian submanifolds in ℂPn(4) and in ℂHn(− 4) which satisfy this basic equality. In this article, we obtain the explicit expressions of these Lagrangian immersions. As an application, we obtain new Lagrangian immersions of the topological n-sphere into ℂPn(4) and ℂHn(−4).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

[1]Bolton, J., Jensen, G. R., Rigoli, M. and Woodward, L. M.. On conformal minimal immersions of S 2 into ℂPn. Math. Ann. 279 (1988), 599620.CrossRefGoogle Scholar
[2]Chen, B. Y.. Total mean curvature and submanifolds of finite type (World Scientific, 1984).CrossRefGoogle Scholar
[3]Chen, B. Y.. Jacobi's elliptic functions and Lagrangian immersions. Proc. Royal Soc. Edinburgh Sect. A, Math. 126 (1996) (to appear).Google Scholar
[4]Chen, B. Y., Dillen, F., Verstraelen, L. and Vrancken, L.. An exotic totally real minimal immersion of S 3 in ℂP3 and its characterization. Proc. Royal Soc. Edinburgh Sect. A, Math. 126 (1996) (in press).Google Scholar
[5]Chen, B. Y., Ludden, G. D. and Montiel, S.. Real submanifolds of a Kaehlerian manifold. Algebra, Groups and Geometries 1 (1984), 176212.Google Scholar
[6]Lawden, D. F., Elliptic functions and applications (Springer–Verlag, 1989).CrossRefGoogle Scholar
[7]Reckziegel, H.. Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion; in Global differential geom and global analysis (1984). Lecture Notes in Mathematics (Springer-Verlag, 1985), pp. 264279.Google Scholar

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