It is known that if $M$ is a complete, connected Riemannian manifold of constant sectional curvature, then its Riemannian universal cover has to be isometric to one of the three models spaces: Euclidean space, the n-dimensional sphere, or hyperbolic n-space. However, I cannot seem to find anything in the literature regarding what can we say if the base space does not have constant sectional curvature.
For example, even if $M$ had (non-constant) everywhere negative sectional curvature, can we really say that its Riemannian universal covering is still isometric to $\mathbb{H}^n$? Although this seems to be the only reasonable answer, the action of $\pi_1(M)$ on the universal cover would yield a constant curvature metric on $M$, which we do not have by assumption, so I am a bit confused on this point.
Any references or a brief explanation would be much appreciated.