The twistor sections on the Wolf spaces
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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 $\|s\|^2$ 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, $\|s\|^2$ is a Morse function in the sense of Bott and its critical manifolds consist of the zero locus and $S_M$.References
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Additional Information
- Yasuyuki Nagatomo
- Affiliation: Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka 810-8560, Japan
- Email: nagatomo@math.kyushu-u.ac.jp
- Received by editor(s): February 16, 2004
- Published electronically: April 4, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 4497-4517
- MSC (2000): Primary 53C26
- DOI: https://doi.org/10.1090/S0002-9947-08-04552-2
- MathSciNet review: 2403694