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We evaluate the Wightman function, the mean field squared and the vacuumexpectation value (VEV) of the energy-momentum tensor for a scalar field withRobin boundary condition on a spherical shell in the background of a constantnegative curvature space. For the coefficient in the boundary condition thereis a critical value above which the scalar vacuum becomes unstable. In bothinterior and exterior regions, the VEVs are decomposed into the boundary-freeand sphere-induced contributions. For the latter, rapidly convergent integralrepresentations are provided. In the region inside the sphere, the eigenvaluesare expressed in terms of the zeros of the combination of the associatedLegendre function and its derivative and the decomposition is achieved bymaking use of the Abel-Plana type summation formula for the series over thesezeros. The sphere-induced contribution to the VEV of the field squared isnegative for Dirichlet boundary condition and positive for Neumann one. Atdistances from the sphere larger than the curvature scale of the backgroundspace the suppression of the vacuum fluctuations in the gravitational fieldcorresponding to the negative curvature space is stronger compared with thecase of the Minkowskian bulk. In particular, the decay of the VEVs with thedistance is exponential for both massive and massless fields. The correspondingresults are generalized for spaces with spherical bubbles and for cosmologicalmodels with negative curvature spaces.

Wightman function and the Casimir effect for a Robin sphere in a constant curvature space