Transversally Complex Submanifolds of a Quaternion Projective Space

Abstract

We study a kind of complex submanifolds in a quaternion projective space $\mathbb {H}P^n$, which we call transversally complex submanifolds, from the viewpoint of quaternionic differential geometry. There are several examples of transversally complex immersions of Hermitian symmetric spaces. For a transversally complex immersion $f:M\rightarrow \mathbb {H}P^n$, a key notion is a Gauss map associated with f, which is a map $S:M \rightarrow \mathrm{End}(\mathbb {H}^{n+1})$ with $S^2 = -\mathrm{id}$. Our theory is an attempt of a generalization of the theory “Conformal geometry of surfaces in $S^4$ and quaternions” by Burstall, Ferus, Leschke, Pedit, and Pinkall [4].