We study the persistence P(t) of the magnetization of a d' dimensional
manifold (i.e., the probability that the manifold magnetization does not flip
up to time t, starting from a random initial condition) in a d-dimensional spin
system at its critical point. We show analytically that there are three
distinct late time decay forms for P(t) : exponential, stretched exponential
and power law, depending on a single parameter \zeta=(D-2+\eta)/z where D=d-d'
and \eta, z are standard critical exponents. In particular, our theory predicts
that the persistence of a line magnetization decays as a power law in the d=2
Ising model at its critical point. For the d=3 critical Ising model, the
persistence of the plane magnetization decays as a power law, while that of a
line magnetization decays as a stretched exponential. Numerical results are
consistent with these analytical predictions.Comment: 4 pages revtex, 1 eps figure include