Almost-Hermitian Structures, Twistor Spaces and Harmonic Maps

Abstract

In the first part of the paper, we review the notion of an almost-complex structure and some topological obstructions for existence of such a structure. Also, we present basic facts about twistor spaces as parametrization spaces of the almostHermitian structures, i.e. orthogonal almost-complex structures, on a Riemannian (or conformal) manifold. In the second part, we consider the problem of when an almost-Hermitian structure on a Riemannian or conformal manifold determines a harmonic or pseudo-harmonic map from the manifold into its twistor space. Recent results on this problem by the author (and collaborators) are discussed mainly in the case of a four-dimensional base manifold. Since this survey is intended for a wide audience, including students, in both parts, we emphasize mostly on elucidating examples rather than technicalities of the proofs. 2020 Mathematics Subject Classification: 53C43, 58E20, 53C15.