The Gauss maps of transversally complex submanifolds of a quaternion projective space

Abstract

We study a kind of complex submanifolds in a quaternion projective space, which we call transversally complex submanifolds, from the viewpoint of quaternionic differential geometry. We treat them applying the theory of the quaternionic vector bundles. For a transversally complex immersion, we define a Gauss map whose values are complex structures of a quaternionic vector space. It is a generalization of “the mean curvature sphere” in the theory by Burstall, Ferus, Leschke, Pedit, and Pinkall. The Gauss map is a key notion for our theory. As an application, we show a characterization of complex projective spaces which are transversally complex submanifolds of a quaternion projective space.