  We show that the energy differences between the lowest optical singlet
exciton and the lowest triplet exciton in semiconducting single-walled carbon
nanotubes with diameter $\sim 1$ nm and graphene nanoribbons with widths $\sim
2$ nm are an order of magnitude smaller than in the $\pi$-conjugated polymer
poly(para-phenylenevinylene). Our calculated energy gaps between the singlet
and triplet excitons are in excellent agreement with the measured values in
three different nanotubes with diameters close to 1 nm. The spatial extent of
the triplet exciton is nearly the same as that of the singlet exciton in wide
nanotubes and nanoribbons, in contrast to that in $\pi$-conjugated polymers, in
which the triplet exciton exhibits strong spatial confinement. Weakly confined
behavior of the triplet state begins in nanoribbons with widths as narrow as
2.5 times the graphene unit lattice vector. We discuss possible consequences of
the small singlet-triplet energy difference in the carbon nanostructures on
device applications.
