I was wondering that, if I have a manifold $M=\mathbb{H}^n / \Gamma$ (where $\Gamma$ is a discrete group of isometries of the hyperbolic space), how would the curvature elements of this manifold change? More concretely, I mean if the Riemann or Ricci tensors coincide with those of $\mathbb{H}^n$.
My intuition says no, since both tensors are locally defined and they are just a function of the metric for the Levi Civita connection.
Thank you so much in advance!!