3-Sasakian manifolds in dimension seven, their spinors and [formula omitted]-structures

Abstract

It is well known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G 2 -structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G 2 -structure in this family whose characteristic connection ∇ c along with its parallel spinor field Ψ 0 can be used for a thorough investigation of the geometric properties of 7-dimensional 3-Sasakian manifolds. Many known and some new properties can be easily derived from the properties of ∇ c and of Ψ 0 , yielding thus an appropriate substitute for the missing characteristic connection.