2000/07/29 by Charles P. Boyer, Krzysztof Galicki, Paolo Piccinni
Mathematics · #math.DG
published as Annals of Global Analysis and Geometry 21, 85-110, 2002. · 22 pages
arxiv created 2000/07/29 · arxiv updated 2009/11/30
Using 3-Sasakian reduction techniques we obtain infinite families of new 3-Sasakian manifolds \scriptstyle\cal M(p1,p2,p3) and \scriptstyle\cal M(p1,p2,p3,p4) in dimension 11 and 15 respectively. The metric cone on \scriptstyle\cal M(p1,p2,p3) is a generalization of the Kronheimer hyperkähler metric on the regular maximal nilpotent orbit of \scriptstyle\Got s\Got l(3,\bbc) whereas the cone on \scriptstyle\cal M(p1,p2,p3,p4) generalizes the hyperkähler metric on the 16-dimensional orbit of \scriptstyle\Got s\Got o(6,\bbc). These are first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further \scriptstyleU(1)-reductions of \scriptstyle\cal M(p1,p2,p3). These yield examples of non-toric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families \scriptstyle\cal O(Θ) of compact self-dual positive scalar curvature Einstein metrics with orbifold singularities and with only one Killing vector field.