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

2017/11/01 by Kim, Inyoung
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.00173

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 \mathbbCP₂ with a slightly weaker curvature hypothesis and there is an infinite dimensional moduli space of such metrics near the Fubini-Study metric on \mathbbCP₂. We discuss some of conditions which can be added in order to get the Fubini-Study metric up to diffeomorphisms and rescaling.

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