  We study global properties of the global (center-)stable manifold of a
normally attracting invariant manifold (NAIM), the special case of a normally
hyperbolic invariant manifold (NHIM) with empty unstable bundle. We restrict
our attention to continuous-time dynamical systems, or flows. We show that the
global stable foliation of a NAIM has the structure of a topological disk
bundle, and that similar statements hold for inflowing NAIMs and for general
compact NHIMs. Furthermore, the global stable foliation has a $C^k$ disk bundle
structure if the local stable foliation is assumed $C^k$. We then show that the
dynamics restricted to the stable manifold of a compact inflowing NAIM are
globally topologically conjugate to the linearized transverse dynamics at the
NAIM. Moreover, we give conditions ensuring the existence of a global $C^k$
linearizing conjugacy. We also prove a $C^k$ global linearization result for
inflowing NAIMs; we believe that even the local version of this result is new,
and may be useful in applications to slow-fast systems. We illustrate the
theory by giving applications to geometric singular perturbation theory in the
case of an attracting critical manifold: we show that the domain of the
Fenichel Normal Form can be extended to the entire global stable manifold, and
under additional nonresonance assumptions we derive a smooth global linear
normal form.
