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I have seen a lot of discussion of alternate geometries for example on a sphere or hyperbolic saddle as opposed to a plane:

Has anyone consider the notion of that plane or hyperbolic saddle itself being embedded on a 4-sphere or 4-saddle and once again that 4-saddle itself embedded on a higher dimensional analog?

What effect does this have on geometry locally?

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My question is really whether there exists a general theory for this – frogeyedpeas May 7 '13 at 3:25

Great question!

There actually is a general theory for this, or more accurately, many general theories. The different theories depend on what feature you want to generalize. Here are some examples, though I am sure the list is far from complete:

  • Riemannian manifolds: This is a huge area with much work in it! Here, you say that the key feature of these spherical and hyperbolic geometries is the existence of an infinitesimal notion of distance and angle. In this setting, you can consider, for instance, a space with a very "bumpy" surface.

  • Homogeneous spaces: a very special class of Riemannian manifold is one for which every point looks the same -- this is true for the sphere and the hyperbolic space. A lot can be said about these, and there is much work in classifying these and seeing what you get if you loosen these conditions up slightly.

  • Constant curvature, negative curvature, constant mean curvature, etc. : these take some of the features of the sphere and of the hyperbolic plane (constant curvature) and interpret what they mean if you are in higher dimensions.

I don't fully understand what you mean by seeing the hyperbolic plane sitting in a higher dimensional hyperbolic space. The geometric properties of these spaces don't use anything extrinsic about them, so I don't see how interpreting them as sitting inside a specific bigger space gives you any more ideas to work with. OTOH, I may have misunderstood what you meant.

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You gave me a good lead on the topic. As for what I meant: the way one can embed a usual triangle in a sphere, or a circle on the surface of a sphere, is it possible that the sphere in question itself is embedded. On the hyper surface of another object. So it's not just a Euclidean sphere but rather a sphere embedded on something else like a higher dimensional saddle or torus etc... – frogeyedpeas May 9 '13 at 16:11
What do you mean by embedding a usual triangle (or circle) in a sphere? In the case of the circle, you can mean that you take the set of points that are the same (spherical) distance from a given point. I don't know off the top of my head what these geometries are called. A Riemannian geometer should be able to answer this. – Sam Lisi May 10 '13 at 11:40

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