Riemannian geometry vs Hyperbolic geometry 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.
 A: Hyperbolic manifolds do arise, as you noted, as a special case of Riemannian manifolds. 
But that does not mean "hyperbolic geometry (as a mathematical subject) is a part of the Riemannian geometry". This is because that since hyperbolic manifolds are special examples of Riemannian manifolds, there are properties of hyperbolic manifolds that are not shared by general Riemannian manifolds. So techniques have been developed to study hyperbolic geometry that have no clear analogue in Riemannian geometry in general. 
If you think about it, you will see that this is a general fact:
If a class of objects $X$ is a subset of another class of objects $Y$, then the class of techniques $\mathcal{X}$ that can be used to study $X$ must include, as a subset, the class of techniques $\mathcal{Y}$ used to study $Y$. 
A: A manifold with (negative,zero, positive) curvatures  within the broader framework of Riemannian geometry is to be understood as being essential part of (hyperbolic,parabolic,elliptic) geometries respectively.
