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SELF-DUAL EINSTEIN HERMITIAN FOUR MANIFOLDS VESTISLAV APOSTOLOV AND PAUL GAUDUCHON
 

Summary: SELF-DUAL EINSTEIN HERMITIAN FOUR MANIFOLDS
VESTISLAV APOSTOLOV AND PAUL GAUDUCHON
Abstract. We provide a local classification of self-dual Einstein Rie-
mannian four manifolds admitting a positively oriented Hermitian struc-
ture and characterize those which carry a hyperhermitian, non-hyperk¨ahler
structure compatible with the negative orientation. We show that self-
dual Einstein 4-manifolds obtained as quaternionic quotients of HP2
and HH2
are Hermitian.
2000 Mathematics Subject Classification. Primary 53B35, 53C55
Keywords: Einstein metrics; complex structures; hypercomplex struc-
tures; quaternionic K¨ahler manifolds.
Introduction
This paper is concerned with oriented, four-dimensional Einstein man-
ifolds which are Hermitian, i.e. admit a positively oriented (integrable)
complex structure, and are self-dual, meaning that the anti-self-dual Weyl
tensor W- vanishes identically.
A Riemannian version of the celebrated Goldberg-Sachs theorem of Gen-
eral Relativity implies that a Riemannian Einstein 4-manifold locally admits
a positively oriented Hermitian structure if and only if the self-dual Weyl

  

Source: Apostolov, Vestislav - Département de mathématiques, Université du Québec à Montréal

 

Collections: Mathematics