  We extend our previous analysis of Riemannian four-manifolds M admitting
rigid supersymmetry to N=1 theories that do not possess a U(1)_R symmetry. With
one exception, we find that M must be a Hermitian manifold. However, the
presence of supersymmetry imposes additional restrictions. For instance, a
supercharge that squares to zero exists, if the canonical bundle of the
Hermitian manifold M admits a nowhere vanishing, holomorphic section. This
requirement can be slightly relaxed if M is a torus bundle over a Riemann
surface, in which case we obtain a supercharge that squares to a complex
Killing vector. We also analyze the conditions for the presence of more than
one supercharge. The exceptional case occurs when M is a warped product S^3 x
R, where the radius of the round S^3 is allowed to vary along R. Such manifolds
admit two supercharges that generate the superalgebra OSp(1|2). If the S^3
smoothly shrinks to zero at two points, we obtain a squashed four-sphere, which
is not a Hermitian manifold.
