  We study the fourth order Schr\"odinger equation with mixed dispersion on an
$N$-dimensional Cartan-Hadamard manifold. At first, we focus on the case of the
hyperbolic space. Using the fact that there exists a Fourier transform on this
space, we prove the existence of a global solution to our equation as well as
scattering for small initial data. Next, we obtain weighted Strichartz
estimates for radial solutions on a large class of rotationally symmetric
manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ.
Equ., 07). Finally, we give a blow-up result for a rotationally symmetric
manifold relying on a localized virial argument.
