On G/H geometry and its use in M-theory compactifications

The Riemannian geometry of coset spaces is reviewed, with emphasis on its applications to supergravity and M-theory compactifications. Formulae for the connection and curvature of rescaled coset manifolds are generalized to the case of nondiagonal Killing metrics. The example of the N010 spaces is discussed in detail. These are a subclass of the coset manifolds Npqr=G/H=SU(3)xU(1)/U(1)xU(1), the integers p, q, r characterizing the embedding of H in G. We study the realization of N010 as G/H=SU(3)xSU(2)/U(1)xSU(2) (with diagonal embedding of the SU(2) is an element of H into G). For a particular G-symmetric rescaling there exist three Killing spinors, implying N=3 supersymmetry in the AdS4xN010 compactification of D=11 supergravity. This rescaled N010 space is of particular interest for the AdS4/CFT3 correspondence, and its SU(3)xSU(2) isometric realization is essential for the OSp(4 vertical bar 3) classification of the Kaluza-Klein modes