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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.

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    $\begingroup$ A hyperbolic manifold is a complete Riemannian manifold of constant sectional curvature $-1$. So I would say, this belongs to Riemannian Geometry. But let us read what the geometers say themselves - see here. $\endgroup$ – Dietrich Burde Jun 18 '15 at 8:41
  • $\begingroup$ @DietrichBurde Good book...@ivo You may find the figure 34 on page 99 helpful. $\endgroup$ – Troy Woo Jun 18 '15 at 8:56
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    $\begingroup$ @TroyWoo Sure, these areas are "highly connected". I know this from number theory (which is also in the picture - automorphic forms). $\endgroup$ – Dietrich Burde Jun 18 '15 at 9:05
  • $\begingroup$ @DietrichBurde You are referring to the Langlands correspondence? $\endgroup$ – Troy Woo Jun 18 '15 at 9:24
  • $\begingroup$ This reminds me of a question I once asked about hyperbolic geometry and spaces with indefinite metric signature which you may find useful. $\endgroup$ – rschwieb Jun 18 '15 at 10:27
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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 $Y$.

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  • $\begingroup$ Do you mind making some comments on Cartan geometry? I never quite get the idea. $\endgroup$ – Troy Woo Jun 18 '15 at 9:05
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    $\begingroup$ In the context of this answer, Cartan geometry allows more objects than Riemannian geometry. You can see Riemannian geometry as a special case with a particular choice of local geometry. So techniques developed for Cartan geometry will definitely apply also for Riemannian geometry but not necessarily vice versa. $\endgroup$ – Willie Wong Jun 18 '15 at 9:46
  • $\begingroup$ Thanks, I guess it is not reasonable to ask you explain Cartan geometry in detail here. $\endgroup$ – Troy Woo Jun 18 '15 at 9:51
  • $\begingroup$ (There's a comment from @user249018 , put incorrectly as an answer): Thanks for the comments. Can you refer to any of the properties and techniques which distinguish the hyperbolic manifolds from the general Riemannian manifolds ? $\endgroup$ – user99914 Jun 18 '15 at 10:21
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    $\begingroup$ @user249108 / OP: what do you / don't you know? The very fact you cited in your question (constant negative curvature) already distinguishes hyperbolic manifolds from general Riemannian ones. Have you looked at standard resources like Wikipedia? But one spectacular example is the fact that the geometry of hyperbolic space admits an axiomatization very similar to, yet different from, that of Euclidean spaces. $\endgroup$ – Willie Wong Jun 18 '15 at 11:25
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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.

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