Rigid supersymmetric theories in 4d Riemannian space

We consider rigid supersymmetric theories in four-dimensional Riemannian spin manifolds. We build the Lagrangian directly in Euclidean signature from the outset, keeping track of potential boundary terms. We reformulate the conditions for supersymmetry as a set of conditions on the torsion classes of a suitable SU(2) or trivial G-structure. We illustrate the formalism with a number of examples including supersymmetric backgrounds with non-vanishing Weyl tensor.