  The Closest Point Method for solving partial differential equations (PDEs)
posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput.
Phys. 2008] and successfully applied to a variety of surface PDEs. In this
paper we study the theoretical foundations of this method. The main idea is
that surface differentials of a surface function can be replaced with Cartesian
differentials of its closest point extension, i.e., its composition with a
closest point function. We introduce a general class of these closest point
functions (a subset of differentiable retractions), show that these are exactly
the functions necessary to satisfy the above idea, and give a geometric
characterization of this class. Finally, we construct some closest point
functions and demonstrate their effectiveness numerically on surface PDEs.
