The large-time behavior of solutions to Burgers equation with
small viscosity is described using invariant manifolds. In particular,
a geometric explanation is provided for a phenomenon known as
metastability,which in the present context means that
solutions spend a very long time near the family of solutions known as
diffusive N-waves before finally converging to a stable self-similar
diffusion wave. More precisely, it is shown that in terms of
similarity, or scaling, variables in an algebraically weighted $L^2$
space, the self-similar diffusion waves correspond to a one-dimensional
global center manifold of stationary solutions. Through each of these
fixed points there exists a one-dimensional, global, attractive,
invariant manifold corresponding to the diffusive N-waves. Thus,
metastability corresponds to a fast transient in which solutions
approach this ``metastable" manifold of diffusive N-waves, followed by
a slow decay along this manifold, and, finally, convergence to the
self-similar diffusion wave.