It is known that the metric and Palatini formalisms of gravity theories have
their own interesting features but also suffer from some different drawbacks.
Recently, a novel gravity theory called hybrid metric-Palatini gravity was put
forward to cure or improve their individual deficiencies. The action of this
gravity theory is a hybrid combination of the usual Einstein-Hilbert action and
a $f(\mathcal{R})$ term constructed by the Palatini formalism. Interestingly,
it seems that the existence of a light and long-range scalar field in this
gravity may modify the cosmological and galactic dynamics without conflicting
with the laboratory and Solar System tests. In this paper we focus on the
tensor perturbation of thick branes in this novel gravity theory. We consider
two models as examples, namely, the thick branes constructed by a background
scalar field and by pure gravity. The thick branes in both models have no inner
structure. However, the graviton zero mode in the first model has inner
structure when the parameter in this model is larger than its critical value.
We find that the effective four-dimensional gravity can be reproduced on the
brane for both models. Moreover, the stability of both brane systems against
the tensor perturbation can also be ensured.Comment: 7 pages, 4 figure