An example of non-compact totally complex submanifolds of compact quaternionic K\"{a}hler symmetric spaces

Abstract

Totally complex submanifolds of a quaternionic K\"{a}hler manifold are analogous to complex submanifolds of a K\"{a}hler manifold. In this paper, we construct an example of a non-compact totally complex submanifold of maximal dimension of a compact quaternionic K\"{a}hler symmetric space, except for quaternionic projective spaces. A compact Lie group acts on our example isometrically, and this action is of cohomogeneity one. Our example is a holomorphic line bundle over some Hermitian symmetric space of compact type. Moreover, each fiber is a totally geodesic submanifold of the ambient quaternionic K\"{a}hler symmetric space and our example is a ruled submanifold. Our construction relies on the action of a subgroup of the isometry group and a maximal totally geodesic sphere with maximal sectional curvature known as a Helgason sphere. Furthermore, we prove that there exist no compact submanifolds of the same dimension that contain our example as an open part, except where the ambient quaternionic K\"{a}hler symmetric space is a complex Grassmannian.