  In this short note we prove Lieb--Thirring inequalities on manifolds with
negative constant curvature. The discrete spectrum appears below the continuous
spectrum $(d-1)^2/4, \infty)$, where $d$ is the dimension of the hyperbolic
space. As an application we obtain a P\'olya type inequality with not a sharp
constant. An example of a 2D domain is given for which numerical calculations
suggest that the P\'olya inequality holds for it.
