Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes

A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time

“…A wide generalization are the "generalized Robertson-Walker spacetimes", introduced in 1995 by Alías, Romero and Sánchez [1,2]: Definition 1.1. An n-dimensional Lorentzian manifold is a generalized RobertsonWalker space-time (GRW) if locally the metric may take the form: ds 2 = −dt 2 + q(t) 2 g * αβ (x 2 , .…”

Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes

“…That is, −I × f M n is nothing but a Lorentzian warped product with Lorentzian base (I, −dt 2 ), Riemannian fiber (M n , , M ), and warping function f . Following the terminology introduced in [4], we will refer to −I × f M n as a generalized Robertson-Walker (GRW) spacetime. In particular, when the Riemannian factor M n has constant sectional curvature then −I × f M n is classically called a Robertson-Walker (RW) spacetime.…”

“…Even more, when k is even it follows from the Gauss equation of the hypersurface that the value of H k is a geometric quantity which is related to the intrinsic curvature of Σ. For instance, for k = 2 one gets that (4) n(n − 1)…”

Integral Formulae for Spacelike Hypersurfaces in Conformally Stationary Spacetimes and Applications

“…As for the case of more general Lorentzian ambient spaces, in a series of recent papers, the first author together with Romero and Sánchez [5][6][7] studied the uniqueness of space-like hypersurfaces with constant mean curvature in a wide class of Lorentzian manifolds, the so-called conformally stationary spacetimes. Since our work will be developed in such ambient spacetimes, we will describe here that class of Lorentzian manifolds, and its relation to other known spacetimes, such as generalized Robertson-Walker spacetimes.…”