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An exotic totally real minimal immersion of S3 in ℂP3 and its characterisation

Published online by Cambridge University Press:  14 November 2011

B.-Y. Chen ,
F. Dillen ,
L. Verstraelen  and
L. Vrancken
B.-Y. Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, U.S.A.
F. Dillen
Affiliation:
K.U. Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B-3001 Leuven, Belgium
L. Verstraelen
Affiliation:
K.U. Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B-3001 Leuven, Belgium
L. Vrancken
Affiliation:
K.U. Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B-3001 Leuven, Belgium
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Abstract

In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 1 , 1996 , pp. 153 - 165
Copyright
Copyright © Royal Society of Edinburgh 1996

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