I'm currently reading Ratcliffe's Foundations of Hyperbolic Geometry (third edition). I'm having trouble resolving a possible contradiction in one of the theorems.
Let $M$ be a hyperbolic $3$-manifold obtained by gluing finitely many hyperbolic tetrahedra by isometries. Theorem 10.2.2 states that $M$ is complete if and only if the links of every cusp is complete. However, Theorem 10.2.1 states that the link of a cusp point of $M$ is either a Klein bottle or a torus, both of which are compact.
Since the link is compact, it is therefore a complete metric space. However, this is clearly wrong because not every hyperbolic $3$-manifold is complete. What am I misunderstanding?
This is not about a metric on the torus or Klein bottle per se. Instead, this is about a metric on a torus cusp or on a Klein bottle cusp.
When we talk about a "torus cusp" we mean a closed subset of $M$ homeomorphic to $T^2 \times [0,\infty)$. To say that this cusp is complete means that when you restrict the hyperbolic metric on $M$ to obtain a metric on $T^2 \times [0,\infty)$, the result is a complete metric. Similarly for a Klein bottle cusp.
In fact, the toruses or Klein bottles themselves, i.e. the ones which arise as a Cartesian factor of a cusp, do not have natural metrics in this setting. At best, they have a natural similarity class of metrics; and yes, as you say, any metric in that similarity class is complete because it is defined on a compact space.
Nonetheless, that's not what's going on here. A $T^2 \times [0,\infty)$ cusp is not compact, and so the issue of whether it is complete in its restricted metric is of significant meaning.