  We study travelling wave solutions, that is, solutions of the form $v(t, x) =
e^{i\lambda t}u(g(t)x)$, to nonlinear Schr\"odinger and Klein-Gordon equations
on Riemannian manifolds, both compact and non-compact ones, with emphasis on
the NLKG. Here $g(t)$ represents a one-parameter family of isometries generated
by a Killing field $X$ and a case of particular interest is when $X$ has length
$\leq 1$, which leads in certain settings to hypoelliptic operators with loss
of at least one derivative. In the compact case, we establish existence of
travelling wave solutions via "energy" minimization methods and prove that at
least compact isotropic manifolds have \emph{genuinely} travelling waves. We
establish certain sharp regularity estimates on low dimensional spheres that
improve results in ~\cite{T1} and carry out the subelliptic analysis for NLKG
on spheres of higher dimensions utilizing their homogeneous coset space
properties. Such subelliptic phenomenon have no parallel in the setting of flat
spaces. We will also study related phenomenon on complete noncompact manifolds
with certain symmetry assumptions using concentration-compactness type
arguments.
