We present a simple explicit construction of hyper-Kaehler and hyper-symplectic (also known as neutral hyper-Kaehler or hyper-parakaehler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kaehler and hyper... View more
2.3. The Heisenberg group H3, Bianchi type II, Gibbons-Hawking class. Consider the two-step nilpotent Heisenberg group H3 defined by the structure equations 3.1. Bianchi type IX hyper-symplectic metrics and hyper-K¨ahler Bianchi type V III hyperK¨ahler metrics. Let G = SU (2) = S3 be described by the structure equations (2.10). We evolve the SU (2) structure according to (2.8). Using (2.10), we reduce the evolution equations (3.4) to the already considered system (2.20). This establishes a correspondence between triaxial Bianchi type IX hyper-symplectic metrics and triaxial hyperK¨ahler Bianchi type V III hyper-K¨ahler metrics. The general solution is given by (2.21). Taking f = f1f2f3 in (2.21) and all fi different we obtain explicit expression of a triaxial hyper-symplectic metric (3.6), where the forms e1, e2, e3 and the functions f1, f2, f3, f are given by (2.11) and (2.21), respectively. A particular solution is obtained by letting a1 = a2 = 0, a3 = 1a6 in (2.21) which gives
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