Supersymmetric geometries of IIA supergravity I

IIA supergravity backgrounds preserving one supersymmetry locally admit four types of Killing spinors distinguished by the orbits of Spin(9,1) on the space of spinors. We solve the Killing spinor equations of IIA supergravity with and without cosmological constant for Killing spinors representing two of these orbits, with isotropy groups Spin(7) and Spin(7) ⋉ ℝ⁸. In both cases, we identify the geometry of spacetime and express the fluxes in terms of the geometry. We find that the geometric constraints of backgrounds with a Spin(7) ⋉ ℝ⁸ invariant Killing spinor are identical to those found for heterotic backgrounds preserving one supersymmetry