  The radion on the de Sitter brane is investigated at the linear perturbation
level, using the covariant curvature tensor formalism developed by Shiromizu,
Maeda and Sasaki. It is found that if there is only one de Sitter brane with
positive tension, there is no radion and thus the ordinary Einstein gravity is
recoverd on the brane other than corrections due to the massive Kaluza-Klein
modes. As a by-product of using the covariant curvature tensor formalism, it is
immediately seen that the cosmological scalar, vector and tensor type
perturbations all have the same Kaluza-Klein spectrum. On the other hand, if
there are two branes with positive and negative tensions, the gravity on each
brane takes corrections from the radion mode in addition to the Kaluza-Klein
modes and the radion is found to have a negative mass-squared proportional to
the curvature of the de Sitter brane, in contrast to the flat brane case in
which the radion mass vanishes and degenerates with the 4-dimensional graviton
modes. To relate our result with the metric perturbation approach, we derive
the second order action for the brane displacement. We find that the radion
identified in our approach indeed corresponds to the relative displacement of
the branes in the Randall-Sundrum gauge and describes the scalar curvature
perturbations of the branes in the gaussian normal coordinates around the
branes. Implications to the inflationary brane universe are briefly discussed.
