I'm a bit confused why we always separate the stable/unstable manifold theorem and the central manifold theorem.
The stable/unstable manifold theorem applies to a hyperbolic point ($\mathrm{Re}(\lambda)\neq 0$) and states (roughly) that there is a unique stable manifold and a unique unstable manifold, the dimensions of which corresponds to the dimensions of the stable and unstable manifolds, repectively.
The center manifold states that there is a stable and an unstable manifold, unique, and a (possibly non-unique) center manifold, the dimension of which is the dimension of the center subspace (it of course also states other things which are not of interested here).
Question: Is there additional information is the stable/unstable manifold theorem that is not included in the center manifold theorem, applied to the particular case when the dimension of the central manifold equals to $0$?