IngentaConnect 3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Authors: Boyer C.P.¹; Galicki K.²; Piccinni P.³

Source: Annals of Global Analysis and Geometry, Volume 21, Number 1, March 2002 , pp. 85-110(26)

Publisher: Springer

Abstract:

Using 3-Sasakian reduction techniques we obtain infinite families of new 3-Sasakian manifolds M (p_1, p_2, p_3) and M (p_1, p_2, p_3, p_4) in dimension 11 and 15 respectively. The metric cone on (p_1, p_2, p_3) is a generalization of the Kronheimer hyperkähler metric on the regular maximal nilpotent orbit of sl (3, C) whereas the cone on M (p_1, p_2, p_3, p_4) generalizes the hyperkähler metric on the 16-dimensional orbit of so(6, C). These are the first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further U(1)-reductions of M(p_1, p_2, p_3). These yield examples of nontoric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families O( Theta } of compact self-dual positive scalar curvature Einstein metrics with orbifold singularities and with only one Killing vector field.

Keywords: Sasakian manifolds; Einstein metrics; nilpotent orbits

Language: English

Document Type: Regular paper

Affiliations: 1: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, U.S.A. e-mail: cboyer@math.unm.edu 2: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, U.S.A. e-mail: galicki@math.unm.edu 3: Università degli Studi di Roma, `La Sapienza', Piazzale Aldo Moro 2, I-00185 Rome, Italia. e-mail: piccinni@mat.uniroma1.it

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