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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The twistor sections on the Wolf spaces

Author: Yasuyuki Nagatomo
Journal: Trans. Amer. Math. Soc. 360 (2008), 4497-4517
MSC (2000): Primary 53C26
Published electronically: April 4, 2008
MathSciNet review: 2403694
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Abstract: Let $ M$ be a compact quaternion symmetric space (a Wolf space) and $ V \to M$ an irreducible homogeneous vector bundle on $ M$ with its canonical connection, whose rank is less than or equal to the dimension of $ M$. We classify the zero loci of the transversal twistor sections with a reality condition. There exists a bijection between such zero loci and the real representations of simple compact connected Lie groups with non-trivial principal isotropy subgroups which are neither tori nor discrete groups. Next we obtain an embedding of the Wolf space into a real Grassmannian manifold using twistor sections, which turns out to be a minimal embedding. Finally, we focus our attention on the norm squared $ \Vert s\Vert^2$ of a twistor section $ s$. We identify the subset $ S_M$ where this function attains the maximum value, under a suitable hypothesis. Such sets are classified, and determine totally geodesic submanifolds of the Wolf spaces. Moreover, $ \Vert s\Vert^2$ is a Morse function in the sense of Bott and its critical manifolds consist of the zero locus and $ S_M$.

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Yasuyuki Nagatomo
Affiliation: Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka 810-8560, Japan

Received by editor(s): February 16, 2004
Published electronically: April 4, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.