The $G_2$ geometry of $3$-Sasaki structures

Abstract

We initiate a systematic study of the deformation theory of the second Einstein metric g 1{ ? 5 respectively the proper nearly G 2 structure ϕ 1{ ? 5 of a 3-Sasaki manifold pM 7 , gq. We show that infinitesimal Einstein deformations for g 1{ ? 5 coincide with infinitesimal G 2 deformations for ϕ 1{ ? 5 . The latter are showed to be parametrised by eigenfunctions of the basic Laplacian of g, with eigenvalue twice the Einstein constant of the 4-dimensional base orbifold, via an explicit differential operator. In terms of this parametrisation we determine those infinitesimal G 2 deformations which are unobstructed to second order.

Rigidity of $SU_n$-type symmetric spaces

“…special holonomy) has been taken up by Wang and Wang in [32] and [33] as well as the joint works with Semmelmann [27] [28] . The relationship between deformability of G 2 structures and Einstein metrics is investigated by Nagy and Semmelmann in [22]. 1.5.…”