Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds
  We present a uniform description of $\mathrm{SU}(3)$-structures in dimension $6$ as well as $G_2$-structures in dimension $7$ in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach also enables one to subsume all variations of the notion of a Killing spinor. 
Ilka Agricola, Simon G. Chiossi, Thomas Friedrich, Jos H\"oll
math.DG math-ph math.MP