  In these lecture notes we discuss the solution theory of geometric wave
equations as they arise in Lorentzian geometry: for a normally hyperbolic
differential operator the existence and uniqueness properties of Green
functions and Green operators is discussed including a detailed treatment of
the Cauchy problem on a globally hyperbolic manifold both for the smooth and
finite order setting. As application, the classical Poisson algebra of
polynomial functions on the initial values and the dynamical Poisson algebra
coming from the wave equation are related. The text contains an introduction to
the theory of distributions on manifolds as well as detailed proofs.
