  The answer to this question is no. In fact, in addition to the solution first
obtained by Christiaan Huygens in 1658, given by the cycloid, we show that
there is an infinite number of tautochrone curves. With this goal, we start by
briefly reviewing an the problem of finding out the possible potential energies
that lead to periodic motions of a particle whose period is a given function of
its mechanical energy. There are infinitely many solutions, called sheared
potentials. As an interesting example, we show that a P\"oschl-Teller and the
one-dimensional Morse potentials are sheared relative to one another for
negative energies, clarifying why they share the same periods of oscillations
for their bounded solutions. We then consider periodic motions of a particle
sliding without friction over a track around its minimum under the influence of
an uniform gravitational field. After a brief historical survey of the
tautochrone problem we show that, given the period of oscillations, there is an
infinity of tracks which lead to the same periods. As a bonus, we show that
there are infinitely many tautochrones.
