Rigidity results for Riemannian twistor spaces under vanishing curvature conditions

TL;DR

This paper establishes new rigidity theorems for four-dimensional Riemannian manifolds and their twistor spaces, focusing on curvature conditions like parallel Bochner tensor and self-duality, with classifications and nonexistence results.

Contribution

It proves that ^3 is the only twistor space with parallel Bochner tensor and classifies Hermitian Ricci-parallel and locally symmetric twistor spaces, extending previous results.

Findings

^3 is the unique twistor space with parallel Bochner tensor

Classified Hermitian Ricci-parallel and locally symmetric twistor spaces

Proved nonexistence of conformally flat twistor spaces

Abstract

In this paper we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces.In particular, using the moving frame method, we prove that $\mathbb{CP}^3$ is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold.