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 [52].
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 [5], the total classification of the timelike and spacelike hyperbolic rotation surfaces is given in terms of cmc in 3-dimensional
de Sitter space [S.sup.3.sub.1].
Furthermore, Zheng [9] studied spacelike hypersurfaces in the
de Sitter space. Furthermore, quaternion manifolds are also studied by many mathematicians (see [2], [3], [5]).
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 [1] 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) [21].