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

We initiate a systematic study of the deformation theory of the second Einstein metric g 1{ ? 5 respectively the proper nearly G2 structure φ1{ ? 5 of a 3-Sasaki manifold pM, gq. We show that infinitesimal Einstein deformations for g 1{ ? 5 coincide with infinitesimal G2 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 G2 deformations which are unobstructed to second order. 2000 Mathematics Subject Classification: Primary 53C25, 58H15, 53C10, 58J50, 57R57.

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