  This paper develops and discusses a residual-based a posteriori error
estimate and a space--time adaptive algorithm for solving parabolic surface
partial differential equations on closed stationary surfaces. The full
discretization uses the surface finite element method in space and the backward
Euler method in time. The proposed error indicator bounds the error quantities
globally in space from above and below, and globally in time from above and
locally from below. A space--time adaptive algorithm is proposed using the
derived error indicator. Numerical experiments illustrate and complement the
theory.
