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Lagrangian submanifolds satisfying a basic equality

Published online by Cambridge University Press:  24 October 2008

Bang-Yen Chen  and
Luc Vrancken
Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027 U.S.A. E-mail address: [email protected]
Luc Vrancken
Affiliation:
Departement Wiskunde, Celestijnenlaan 200 B, B-3001 Leuven, Belgium e-mail address: luc.vrancken©wis.kuleuven.ac.be
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 2 , August 1996 , pp. 291 - 307
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

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