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I am reading the Riemannian Geometry, written by Lee, and have just finished the Chapter 3, which ends with The Model Spaces of Riemannian Geometry.

There are three kinds of model spaces $\mathbb R^n$, $\mathbb S^n$ and $\mathbb H^n$. All of their metric can be induced by the pesudo Riemannian metric of Euclidean space, $\mathbb R^n$, $\mathbb S^n$ by the standard metric and $\mathbb H^n$ by the Minkowshi metric.

So my question is why people just study these three kinds of model spaces instead of different Riemannian manifolds, whose metrics induced by the pesudo Riemannian metric of Euclidean space like $$g=(dx^1)^2+\cdots+(dx^r)^2-(dx^{r+1})^2-\cdots-(dx^n)^2$$

Any advice is helpful. Thank you.

gaoxinge
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1 Answers1

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I would suppose that the notion of a model space is just a way to capture the "main features" of (in this case) Riemannian geometry in a more simple and concrete way (concrete meaning they provide good examples,because anyone can imagine how sphere looks and its a bit harder to imagine how some space with some arbitrary metric looks like regardless of being of positive or negative curvature..), so that when you are studying more abstract Riemannian manifolds you can compere them with these more simple spaces and see what features do they share. I believe this model approach first appeared in hyperbolic geometry, where you have manny models for hyperbolic space (like Poincare half-disc model, Klein model,..) which are just variants of the same general hyperbolic space, but made more rigid so they can be applied to various other situations, like Klein model for example has to do with the projective space and so on, well now we know that even this hyperbolic spaces are Riemannian manifolds..

I hope this will help a little although its is not even close to a concrete explanation.

LB1
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