  In this paper, a geometric process to compare solutions of symmetric
hyperbolic systems on (possibly different) globally hyperbolic manifolds is
realized via a family of intertwining operators. By fixing a suitable
parameter, it is shown that the resulting intertwining operator preserves
Hermitian forms naturally defined on the space of homogeneous solutions. As an
application, we investigate the action of the intertwining operators in the
context of algebraic quantum field theory. In particular, we provide a new
geometric proof for the existence of the so-called Hadamard states on globally
hyperbolic manifolds.
