  We give a mathematical construction of Euclidean quantum field theory on
certain curved backgrounds. We focus on generalizing Osterwalder-Schrader
quantization, as these methods have proved useful to establish estimates for
interacting fields on flat space-times. In this picture, a static Killing
vector generates translations in Euclidean time, and physical positivity is
played by positivity under reflection of Euclidean time. We discuss the
quantization of flows which correspond to classical space-time symmetries, and
give a general set of conditions which imply that broad classes of operators in
the classical picture give rise to well-defined operators on the quantum-field
Hilbert space. In particular, Killing fields on spatial sections give rise to
unitary groups on the quantum-field Hilbert space, and corresponding
densely-defined self-adjoint generators. We construct the Schr\"odinger
representation using a method which involves localizing certain integrals over
the full manifold to integrals over a codimension-one submanifold. This method
is called sharp-time localization, and implies reflection positivity.
