  In this note, we study a geometric property of asymptotically Minkowski
spacetimes and an analytic property of the Klein-Gordon operator. Precisely,
our first main results show that asymptotically Minkowski spacetimes are
geodesically complete under a null non-trapping condition. Secondly, we prove
that Sobolev index of a real principal type estimate used in the previous work
is actually optimal.
