  Red giants are evolved stars that have exhausted the supply of hydrogen in
their cores and instead burn hydrogen in a surrounding shell. Once a red giant
is sufficiently evolved, the helium in the core also undergoes fusion.
Outstanding issues in our understanding of red giants include uncertainties in
the amount of mass lost at the surface before helium ignition and the amount of
internal mixing from rotation and other processes. Progress is hampered by our
inability to distinguish between red giants burning helium in the core and
those still only burning hydrogen in a shell. Asteroseismology offers a way
forward, being a powerful tool for probing the internal structures of stars
using their natural oscillation frequencies. Here we report observations of
gravity-mode period spacings in red giants that permit a distinction between
evolutionary stages to be made. We use high-precision photometry obtained with
the Kepler spacecraft over more than a year to measure oscillations in several
hundred red giants. We find many stars whose dipole modes show sequences with
approximately regular period spacings. These stars fall into two clear groups,
allowing us to distinguish unambiguously between hydrogen-shell-burning stars
(period spacing mostly about 50 seconds) and those that are also burning helium
(period spacing about 100 to 300 seconds).
