Calculus of Variations and Geometric Measure Theory

G. Catino - P. Mastrolia - D. Dameno

Rigidity results for Riemannian twistor spaces under vanishing curvature conditions

created by catino on 29 Jun 2022
modified on 15 Jan 2023


Accepted Paper

Inserted: 29 jun 2022
Last Updated: 15 jan 2023

Journal: Ann. Glob. Anal. Geom.
Year: 2023


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.