The twistor sections on the Wolf spaces

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 $S_M$.