Extrinsic hyperspheres in manifolds with special holonomy

TL;DR

This paper investigates the properties of extrinsic hyperspheres in manifolds with special holonomy, proving their nonexistence in quaternion-Kaehler manifolds and introducing a new classification-based approach.

Contribution

It provides a novel classification method for extrinsic hyperspheres using special Killing forms and establishes nonexistence results in quaternion-Kaehler manifolds.

Findings

Extrinsic hyperspheres do not exist in quaternion-Kaehler manifolds.

A new approach based on special Killing forms is developed.

The paper characterizes totally geodesic hypersurfaces in special holonomy manifolds.

Abstract

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.