  In this paper we introduce a notion of scattering theory for the
Laplace-Beltrami operator on non-compact, connected and complete Riemannian
manifolds. A principal condition is given by a certain positive lower bound of
the second fundamental form of angular submanifolds at infinity. Another
condition is certain bounds of derivatives up to order one of the trace of this
quantity. These conditions are shown to be optimal for existence and
completeness of a wave operator. Our theory does not involve prescribed
asymptotic behaviour of the metric at infinity (like asymptotic Euclidean or
hyperbolic metrics studied previously in the literature). A consequence of the
theory is spectral theory for the Laplace-Beltrami operator including
identification of the continuous spectrum and absence of singular continuous
spectrum.
