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 → M an irreducible homogeneous vector bundle onM 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 theWolf 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 SM 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 SM.

Original languageEnglish
Pages (from-to)4497-4517
Number of pages21
JournalTransactions of the American Mathematical Society
Volume360
Issue number9
DOIs
Publication statusPublished - 1 Sep 2008

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