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

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.

References in corpus (1)

• Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, and Almost-Kaehler Geometry