I hope someone can help me or give a ref. I'm trying to understand the general way in which one defines a hyperbolic metric on a given surface $\Sigma$ ( and, if not too complicated, on a manifold of dimension 3-or-higher ).
For a surface $\Sigma$, I guess we can start by using the uniformization theorem to express $\Sigma$ as a quotient of one of the types $D^2/G , \mathbb C/G $ or $S^2/G$ , where $D^2$, $\mathbb C$, $S^2$ are the Poincare disk, the complex plane and the Riemann sphere respectively, and $G$ is a group and the action is transitive and discontinuous.
I can see ( at least I think so ) that if $\Sigma=D^2/G$ , then I think the quotient is a covering map and a local isometry so that we can show that there is a hyperbolic metric $g$ in $\Sigma=D^2/G$ that descends (in each "chart" $U_i$ in the covering map, i.e., in the open sets $U_i$, so that $p: P^{-1}(U_i) \rightarrow U_i$ is a homeomorphism/isometry) to the hyperbolic metric $g$in $D^2$ ( or $g|_{p^{-1}(U_i)}$); i.e., we put together the metrics $g_i$ that descend from $U_i$ to $p^{-1}(U_i)$into a global metric (I guess partitions of unity allow us to glue the whole thing together) . But I cannot see how to do this in other cases where we cannot (or, I should say I don't know how express) $\Sigma$ as a quotient of $D^2$.
Does this method work? Thanks.
A hyperbolic metric on n-dimensional smooth manifold M is a Riemannian metric g in M which is locally isometric to the standard Riemannian metric on the hyperbolic n-space $H^n$. Locally isometric here means that every point in M admits a neighborhood U so that the restriction of g to U is isometric to an open subset of the hyperbolic n-space. Such metric g need not be isometric to a metric of the form $H^n/G$ where G is a discrete group of isometries acting freely on $H^n$. However if M is connected and g is complete, then g is isometric to a quotient metric as above.
Equivalently, one can define hyperbolic metric in M as a Riemannian metric of sectional curvature $-1$, if you know what it means. Do Carmo's book "Riemannian geometry" is a good source for this.
Edit: Here are more details on constructions of hyperbolic metrics, since this is what you are apparently interested in. First, of all, as any Riemannian metric, a hyperbolic Riemannian metric $g$ can be either complete or incomplete. The former (in the hyperbolic setting) are precisely the metrics of the form $$ (M,g)\cong H^n/G, $$ where $G$ is a subgroup of isometries of hyperbolic n-space acting properly discontinuously and freely on $H^n$. Incomplete metrics are, in general, frowned upon: They provide very little insight into structure of $M$. Examples of incomplete hyperbolic metrics are obtained by taking a (noncompact) n-dimensional manifold $M$ which admits an immersion $f: M\to H^n$ (which is the same as an immersion in $R^n$) and then taking pull-back of the hyperbolic metric $h$ on $H^n$: $$ g:= f^*(h). $$ For instance, if $M$ is connected, noncompact and has trivial tangent bundle, it always admits such an immersion and, hence, an incomplete hyperbolic metric.
Metrics on compact manifolds are always complete.
There are several ways to construct complete hyperbolic metrics, as far as I know, they are divided in three types (none of which involves a direct construction of an atlas of charts of local isometries to $H^n$):
(a) Analytical. Examples of such are given by the uniformization theorem for Riemann surfaces, which states that (with exception of $S^2, {\mathbb C}, {\mathbb C}^\times$ and $T^2$) every connected Riemann surface $S$ is conformally equivalent to the quotient $H^2/G$, where $G$ acts isometrically, properly discontinuously and freely on $H^2$. In particular, except for $S^2$ and $T^2$, all connected orientable surfaces admit complete hyperbolic metrics.
Another example is the Ricci flow, which (sometimes) converts a randomly chosen Riemannian metric on a 3-dimensional manifold $M$ to a complete hyperbolic metric on $M$.
(b) Arithmetic. One starts, for instance, with a suitable quadratic form $$ q(x)= a_0 x_0^2 - a_1 x_1^2 - ... - a_n x_n^2 $$
of signature $(1,n)$ and the constants $a_i$ from some number field $F$ with the ring of integers $R$. Then, under some conditions on $a_i$'s, the group $G=Aut(q, R)$ is a discrete subgroup of $Aut(q, {\mathbb R})\cong O(1,n)$ and then one can form the quotient $H^n/G$ (one frequently has to deal with elements of finite order in $G$ though, this is again typically done via some algebraic constructions involving congruence-subgroups).
(c) Geometric and topological: Direct constructions of discrete subgroups of isometries of $H^n$ using fundamental domains or other tricks. There are books by Ratcliffe "Foundations of hyperbolic manifolds" and by Maskit "Kleinian groups" where such constructions are described.
Another geometric construction which is frequently used is by gluing existing hyperbolic manifolds via isometries. For instance, hyperbolic surfaces are frequently described by gluing isometrically "pairs of pants" along their boundary curves. Some people also construct hyperbolic metrics by gluing together ideal hyperbolic simplices.
See also this wikipedia article, especially for references.