Let M be a pseudo-Riemannian spin manifold of dimension n and signature s and denote by N the rank of the real spinor bundle. We prove that M is locally homogeneous if it admits more than 3/4N
Nearby in the stack
independent Killing spinors with the same Killing number, unless
n≡1(mod4)
and
s≡3(mod4)
. We also prove that
M
is locally homogeneous if it admits
k+
independent Killing spinors with Killing number
λ
and
k−
independent Killing spinors with Killing number
−λ
such that
k++k−>3/2N
, unless
n≡s≡3(mod4)
. Similarly, a pseudo-Riemannian manifold with more than
3/4N
independent conformal Killing spinors is conformally locally homogeneous. For (positive or negative) definite metrics, the bounds
3/4N
and
3/2N
in the above results can be relaxed to
1/2N
and
N
, respectively. Furthermore, we prove that a pseudo-Riemannnian spin manifold with more than
3/4N
parallel spinors is flat and that
1/4N
parallel spinors suffice if the metric is definite. Similarly, a Riemannnian spin manifold with more than
3/8N
Killing spinors with the Killing number
λ∈
has constant curvature
4λ2
. For Lorentzian or negative definite metrics the same is true with the bound
1/2N
. Finally, we give a classification of (not necessarily complete) Riemannian manifolds admitting Killing spinors, which provides an inductive construction of such manifolds.