I am learning differential geometry in this semester. Concerning the riemannian geometry, if the cross-sectional curvature (riemannian metric ) is negative at every point, the manifold which arises is hyperbolic. At the other hand hyperbolic geometry is another form of non-euclidean geometry just like the riemannian geometry.
I am wondering if a manifold with negative curvature in the framework of the riemannian geometry is to be understood as being part of hyperbolic geometry ?
If the answer is affirmative, does it mean that hyperbolic geometry is part of the Riemannian geometry ?
If the answer is negative, can one study hyperbolic geometry in the framework of differential manifolds ?
Thanks for your comment.