  We investigate the Wightman function, the vacuum expectation values of the
field squared and the energy-momentum tensor for a massive scalar field with
general curvature coupling parameter subject to Robin boundary conditions on
two codimension one parallel branes located on $(D+1)$-dimensional background
spacetime $AdS_{D_{1}+1}\times \Sigma $ with a warped internal space $\Sigma $.
The general case of different Robin coefficients on separate branes is
considered. Unlike to the purely AdS bulk, the vacuum expectation values
induced by a single brane, in addition to the distance from the brane, depends
also on the position of the brane in the bulk. The brane induced parts in these
expectation values vanish when the brane position tends to the AdS horizon or
AdS boundary. For strong gravitational fields corresponding to large values of
the AdS energy scale, the both single brane and interference parts of the
expectation values integrated over the internal space are exponentially
suppressed. An application to the higher dimensional generalization of the
Randall-Sundrum brane model with arbitrary mass terms on the branes is
discussed. For large distances between the branes the induced surface densities
give rise to an exponentially suppressed cosmological constant on the brane.
