Abstract
The role of some causal symmetries of spacetime which naturally arise in General Relativity is discussed. The importance of spacelike hypersurfaces of constant mean curvature (CMC) in the study of the Einstein equation is recalled. In certain spacetimes with symmetry defined by a timelike gradient conformal vector field or by a lightlike parallel vector field, uniqueness theorems of complete CMC spacelike hypersurfaces are given. In several cases, results of Calabi–Bernstein type are obtained as an application.
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