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I am currently studying The Princeton companion to mathematics.

According the book, "A more precise definition is that the open unit disk is the set of all points $(x,y)$ such that $ x^2 + y^2 < 1$ and that the Riemannian metric on this disk is given by the expression $(dx^2+dy^2)/(1−x^2−y^2)$. This is how we define the square of the distance between $(x,y)$ and $(x + dx,y + dy)$." If the Riemann metric for the Poincare disk is just directly defined as $(dx^2+dy^2)/(1−x^2−y^2)$ without any proof, why can't we randomly use any expression as the Riemann metric? Is there any reason for choosing this specific notion of distance?

Also can there be multiple Riemannian metrics for a given manifold? How can there be multiple metrics for the same manifold?

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  • $\begingroup$ This en.wikipedia.org/wiki/Poincar%C3%A9_disk_model might help. The thing is that Poincare disk is modelled as a hyperbolic space, and the metric you are talking about enforces a negative constant curvature -1 (hence, hyperbolic space). $\endgroup$
    – Tomas
    Commented Mar 23, 2015 at 18:24
  • $\begingroup$ The Poincaré disk model is hyperbolic space, under the right identifications (isometries). $\endgroup$ Commented Mar 23, 2015 at 23:43

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Yes, given an underlying smooth manifold, there are generally many choices of metric. For instance, $dx^2 + dy^2$ (the standard Euclidean metric) is another perfectly acceptable choice. In this sense, there is no single "correct" metric to put on a particular smooth manifold, and it is just a definition.

However, if you give the "same" smooth manifold two different metrics, the results will probably not be the same as Riemannian manifolds. That is, while they have the same smooth structure (they are diffeomorphic), in terms of properties of the metric, they are different (they are not isometric in general).

Now, that said, we might arrive at the same Riemannian manifold via two different constructions. That is, we might start with $M_1$, give it a metric $g_1$, consider a distinct manifold $M_2$, give it some other metric $g_2$, and then show that $(M_1,g_1)$ is isometric to $(M_2,g_2)$.

This is useful, because you may be able to establish facts and properties about one model more easily than you could with the other. In the case of hyperbolic space, one such realization is the Poincare disc, but one can also construct the hyperbolic plane by, e.g., starting with a half-plane (instead of a disc) and a particular metric (and of course there are even more ways to construct this manifold that I won't list here, but look here if you are interested).

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  • $\begingroup$ From what I have read, I think that metric gives the distance between 2 points. So how can there be multiple distances between 2 points? Does each metric give the distance between 2 points along different lines? $\endgroup$
    – Akshit
    Commented Mar 23, 2015 at 18:41
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    $\begingroup$ Each metric gives the distance between 2 points in a different metric. The path that realizes that distance might be the same, it might be different. As a metaphor, consider a map of your city streets as a manifold. We could measure the distance "as the crow flies" between any two points, but that might be of limited use. We could put a different metric that measures "minimum travel-time in a car between point A and B" that is going to give different values of "distance", and is probably going to use a different path (other than the straight line) to connect A and B. $\endgroup$
    – BaronVT
    Commented Mar 23, 2015 at 18:55
  • $\begingroup$ We might further consider "minimum travel-time on the bus between point A and B" which is going to give us yet another set of values, and is going to coincide with the optimal paths for cars in some cases, but going to deviate in others. You could additionally calculate things in your "walking metric" or "cycling metric" etc. $\endgroup$
    – BaronVT
    Commented Mar 23, 2015 at 18:56

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