4-dimensional Riemannian manifolds with a harmonic 2-form of constant length

TL;DR

This paper characterizes 4-dimensional Riemannian manifolds with harmonic 2-forms of constant length, showing they are diffeomorphic to complex projective space and exploring the moduli space of such metrics.

Contribution

It proves that such manifolds are diffeomorphic to P2 under weaker curvature conditions and identifies an infinite-dimensional moduli space near the Fubini-Study metric.

Findings

Manifolds are diffeomorphic to P2.

Existence of an infinite-dimensional moduli space of these metrics.

Conditions to recover the Fubini-Study metric up to diffeomorphism.

Abstract

It was shown by Seaman that if a compact, oriented 4-dimensional riemannian manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, its intersection form is definite and such a harmonic form is unique up to constant multiples. In this paper, we show that such a manifold is diffeomorphic to $\mathbb{CP}_{2}$ with a slightly weaker curvature hypothesis and there is an infinite dimensional moduli space of such metrics near the Fubini-Study metric on $\mathbb{CP}_{2}$. We discuss some of conditions which can be added in order to get the Fubini-Study metric up to diffeomorphisms and rescaling.