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Hyperkähler Structures and Group Actions
Author(s) -
Bielawski Roger
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610796004723
Subject(s) - holonomy , group (periodic table) , ricci curvature , hyperkähler manifold , manifold (fluid mechanics) , pure mathematics , mathematics , metric (unit) , einstein , mathematical physics , physics , mathematical analysis , hermitian manifold , geometry , quantum mechanics , engineering , mechanical engineering , operations management , curvature
A 4 n ‐dimensional Riemannian manifold ( M , g ) is hyperkähler if it possesses three anti‐commuting complex structures I , J , K such that the metric g is Kähler with respect to each of them. The reduced holonomy group of such a manifold is necessarily a subgroup of Sp( n ) so the Ricci tensor of g vanishes and ( M , g ) can be regarded as a positive definite solution to Einstein's equations in vacuum.