  We study classical, real Klein-Gordon theory on Lorentzian Anti de Sitter
(AdS_{1,d}) spacetimes with spatial dimension d. We give a complete list of
well defined and bounded Klein-Gordon solutions for three types of regions on
AdS: slice (time interval times all of space), rod hypercylinder (all of time
times solid ball in space), and tube hypercylinder (all of time times solid
shell in space). Hypercylinder regions are of natural interest for AdS since
the neighborhood of the AdS-boundary is a tube. For the solution spaces of our
regions we find the actions induced by the AdS isometry group SO(2,d). For all
three regions we find one-to-one correspondences between initial data and
solutions on the regions. For rod and tube regions this initial data can also
be given on the AdS boundary. We calculate symplectic structures associated to
the solution spaces, and show their invariance under the isometry actions. We
compare our results to the corresponding expressions for (3+1)-dimensional
Minkowski spacetime, arising from AdS_{1,3} in the limit of large curvature
radius.
