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If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric?

I know that in this case the universal cover is the hyperbolic plane and it has a complete metric. Do we project this metric to the puntured surface? If so, why is it complete?

I will deeply appreciate if somebody gives an example or a good reference.

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2 Answers 2

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A good reference for this is, say, Kobayashi and Nomizu "Foundations of Differential Geometry". The result you are looking for is: If $M$ is a complete Riemannian manifold and $p: M\to M'$ is a (locally) isometric covering map to another Riemannian manifold, then $M'$ is also complete. To prove this note that every geodesic in $M'$ lifts to a geodesic in $M$. Since geodesics in $M$ extend to bi-infinite geodesics, you conclude that $M'$ is geodesically complete. Now, use Hopf-Rinow Theorem. Actually, several converse statements to this are also true, e.g., if both $M, M'$ are complete and $p: M\to M'$ is a locally-isometric map, then $p$ is a covering map. A proof of this is a bit more difficult.

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To see how you get complete metrics, you should know enough about hyperbolic geometry to know what an ideal triangle is. Once you do, note that any triangulation can be made of ideal triangles (usually in more than one way). Checking completeness is then a mildly challenging exercise. I would suggest you look at Bill Thurston's book, as well as Beardon's "Geometry of Discrete groups", and S. Katok's little book (as suggested elsewhere by @Sue).

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