Harmonic Almost Hermitian Structures

Abstract

This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, special attention is paid to the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space of an oriented Riemannian four-manifold. JOHANN DAVIDOVIn general, these critical points are not harmonic maps, but, by analogy, in [36] they are referred to as "harmonic almost complex structures". They are also called "harmonic sections" [35], a term, which is more appropriate in the context of this article.Forgetting the bundle structure of T, we can consider almost Hermitian structures that are critical points of the energy functional under variations through all maps N → T. These structures are genuine harmonic maps from (N, h) into (T, h 1 ) and we refer to [15] for basic fact about such maps.The main goal of this paper is to survey results about the harmonicity (in both senses) of the Atiyah-Hitchin-Singer and Eells-Salamon almost Hermitian structures on the twistor space of an oriented four-dimensional Riemannian manifold, as well as almost Hermitian structures on such a manifold.In Section 2 we recall some basic facts about the twistor spaces of even-dimensional Riemannian manifolds. Special attention is paid to the twistor spaces of oriented four-dimensional manifolds. In Sections 3 and 4 we discuss the energy functional on sections of a twistor space, i.e. almost Hermitian structures on the base Riemannian manifold. We state the Euler-Lagrange equation for such a structure to be a critical point of the energy functional (a harmonic section) obtained by Wood [35,36]. Several examples of non-Kähler almost Hermitian structures, which are harmonic sections are given. Kähler structures are absolute minima of the energy functional. G. Bor, L. Hernández-Lamoneda and M. Salvai [4] have given sufficient conditions for an almost Hermitian structure to be a minimizer of the energy functional. Their result (in fact, part of it) is presented in Section 4 and is used to supply examples of non-Kähler minimizers based on works by C. LeBrun [27] and I. Kim [24]. Section 4 ends with a lemma from [9], which rephrases the Euler-Lagrange equation for an almost Hermitian structure (h, J) on a manifold N in terms of the fundamental 2-form of (h, J) and the curvature of (N, h). This lemma has been used in [9] to show that the Atiyah-Hitchin-Singer almost Hermitian structure J 1 on the negative twistor space Z of an oriented Riemannian 4-manifold (M, g) is a harmonic section if and only if the base manifold (M, g) is self-dual, while the Eells-Salamon structure J 2 is a harmonic section if and only (M, g) is self-dual and of constant scalar curvature. The main part of the proof of this result (slightly different from the proof in [9]) is presented in Section 5. In this context, it is natural to ask when J 1 and J 2 are harmonic maps into the twistor space of Z. Recall that a map between Riem...

Product twistor spaces and Weyl geometry

“…is a local orthonormal frame of Λ 2 ± T M . This frame defines an orientation on Λ 2 ± T M which does not depend on the choice of the frame (E 1 , E 2 , E 3 , E 4 ) (see, for example, [7]). We call this orientation "canonical".…”

Twistorial examples of Riemannian almost product manifolds and their Gil–Medrano and Naveira types

“…is a local orthonormal frame of Λ 2 ± T M defining an orientation on Λ 2 ± T M , which does not depend on the choice of the frame (E 1 , E 2 , E 3 , E 4 ) (see, for example, [6]).…”

“…Then P κ E i = −E i for i = 1, 2 and P κ E j = E j for j = 3, 4. Therefore the dimensions of the (+1) and (−1)-eigenspaces of K 1 , ..., K 4 are (6, 2), (4,4), (2,6), (4,4), respectively. Thus, K ν , ν = 1, ..., 4, are almost product structures on the manifold P.…”

Harmonicity of the Atiyah–Hitchin–Singer and Eells–Salamon almost complex structures

“…In this paper we look at these structures from the point of view of variational theory. The motivation behind is the fact that if a Riemannian manifold admits an almost complex structure compatible with its metric, it possesses many such structures (cf., for example, [6,9]). Thus, it is natural to seek criteria that distinguish some of these structures among all.…”

“…These structures are genuine harmonic maps from (N, h) into (Z, h 1 ); we refer to [12] for basic facts about harmonic maps. The problem when a compatible almost complex structure on a four-dimensional Riemannian manifold is a harmonic map into its twistor space has been studied in [9] (see also [6]).…”