*Chapter Four*

## Poincaré Metrics

In this chapter we consider the formal theory for Poincaré metrics associated to a conformal manifold (*M*, [*g*]). We will see that even Poincaré metrics are in one-to-one correspondence with straight ambient metrics, if both are in normal form. Thus the formal theory for Poincaré metrics is a consequence of the results of Chapter 3. The derivation of a Poincaré metric from an ambient metric was described in [FG], and the inverse construction of an ambient metric as the cone metric over a Poincaré metric was given in §5 of [GrL].

The definition of Poincaré metrics is motivated by the example of the hyperbolic metric 4(1 – *|x|*^{2})^{–2}g_{e} on the ball, where *g*_{e} denotes the Euclidean metric. Let (*M*, [*g*]) be a smooth manifold ...