Maximal totally complex submanifolds of $\mathbb{H}\mathbb{P}^n$: homogeneity and normal holonomy

TL;DR

This paper characterizes maximal totally complex submanifolds of quaternionic projective space, showing they are parallel under certain symmetry or holonomy conditions, advancing understanding of their geometric structure.

Contribution

It proves that such submanifolds are parallel if they are either group orbits or have restricted normal holonomy as a proper subgroup of U(n).

Findings

Maximal totally complex submanifolds are parallel under specified conditions.

Orbit of a compact Lie group implies parallelism.

Restricted normal holonomy being a proper subgroup leads to parallelism.

Abstract

We prove that a maximal totally complex submanifold $N^{2n}$ of the quaternionic projective space $\mathbb{H}\mathbb{P}^n$ ($n\geq 2$) is a parallel submanifold, provided one of the following conditions is satisfied: (1) $N$ is the orbit of a compact Lie group of isometries, (2) the restricted normal holonomy is a proper subgroup of ${\rm U}(n)$.