  We use a one-dimensional random walk on $D$-dimensional hyper-spheres to
determine the critical behavior of statistical systems in hyper-spherical
geometries. First, we demonstrate the properties of such walk by studying the
phase diagram of a percolation problem. We find a line of second and first
order phase transitions separated by a tricritical point. Then, we analyze the
adsorption-desorption transition for a polymer growing near the attractive
boundary of a cylindrical cell membrane. We find that the fraction of adsorbed
monomers on the boundary vanishes exponentially when the adsorption energy
decreases towards its critical value. We observe a crossover phenomenon to an
area of linear growth at energies of the order of the inverse cell radius.
