  The formalism for histories-based generalized quantum mechanics developed in
two earlier papers is applied to the treatment of histories (of particles or
fields or more general objects) in curved spacetimes (which need not admit
foliation in spacelike hypersurfaces). The construction of the space of
temporal supports (a partial semigroup generalizing the space of finite time
sequences employed in traditional temporal description of histories) employs
spacelike subsets of spacetime having dimensionality less than or equal to
three. Definition of symmetry is sharpened by the requirement of continuity of
mappings (employing topological partial semigroups). It is shown that with this
proviso, a symmetry in our formalism implies a conformal isometry of the
spacetime metric.
