But because those publications were focused on the internal constitution of stars, the redshift effect in the de Sitter space
was not emphasized and analysed properly.
This relation allows us to explain the observable effect of the negative mass-square differences for neutrinos by applying de Sitter group for space-time symmetry in the interaction vertex and calculating particles masses as the eigenvalues of the Casimir operator in de Sitter space
Let [M.sup.n + 1.sub.1] (c) be an (n + 1)-dimensional Lorentzian space form with constant sectional curvature c, we separately call it de Sitter space
[S.sup.n+1.sub.1](c), Lorentzian-Minkowski space [L.sup.n+1] or anti-de Sitter space [H.sup.n+1.sub.1] (c), with respect to c > 0, c = 0 or c < 0.
In , the total classification of the timelike and spacelike hyperbolic rotation surfaces is given in terms of cmc in 3-dimensional de Sitter space
Furthermore, Zheng  studied spacelike hypersurfaces in the de Sitter space
. Furthermore, quaternion manifolds are also studied by many mathematicians (see , , ).
Our expanding universe is contained in some sort of wider space (whether a Minkowski space as in the Brout, Englert, Gunzig model or a curved de Sitter space
as in Gott's model) in which the quantum fluctuations occur in the spacetime geometry which 'pull' particles into existence out of the energy locked up in empty space.
Thus the [lambda]-field generating the de Sitter space
(30) is equivalent to the substance described by the energy-impulse tensor
Aiyama  had proved that a compact spacelike submanifold with parallel mean curvature vector and flat normal bundle in de Sitter space
[S.sup.n+p.sub.p](c) is totally umbilical.
That emptiness, Carroll suggests, would be a high-entropy environment technically known as de Sitter space
. "Empty" however, does not convey a precisely correct description.
Witten, "Anti de Sitter space
and holography," Advances in Theoretical and Mathematical Physics, vol.
More specifically, the de Sitter space
whose metric is induced from the pseudo-Euclidean metric (+1, -1, -1, -1, -1) has a specific group of motion which is the pseudo-orthogonal group SO(4,1) .