Partial Classification Results for Positive Quaternion Kaehler Manifolds

TL;DR

This paper proves the conjecture that positive quaternion Kähler manifolds are symmetric spaces in dimension 20 under certain conditions and offers recognition theorems for specific low-dimensional cases.

Contribution

It establishes the symmetry conjecture in a specific dimension and provides recognition criteria for quaternionic projective spaces and the real Grassmannian.

Findings

Proves the symmetry conjecture in dimension 20 under additional assumptions.

Provides recognition theorems for quaternionic projective spaces in low dimensions.

Identifies the real Grassmannian as a positive quaternion Kähler manifold.

Abstract

Positive Quaternion Kaehler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. We prove this conjecture in dimension 20 under additional assumptions and we provide recognition theorems for quaternionic projective spaces (in low dimensions) as well as the real Grassmanian (which is Positive Quaternion Kaehler).