COMPLETE SPACELIKE HYPERSURFACES WITH CONSTANT SCALAR CURVATURE

In this paper, we characterize the n-dimensional (n >= 3) complete spacelike hypersurfaces M-n in a de Sitter space S-1(n+1) with constant scalar curvature and with two distinct principal curvatures one of which is simple. We show that M-n is a locus of moving (n - 1)-dimensional submanifold M-1(n-1) (s), along M-1(n-1) (s) the principal curvature lambda of multiplicity n - 1 is constant and M-1(n-1) (s) is umbilical in S-1(n+1) and is contained in an (n - 1)-dimensional sphere Sn-1 (c(s) = E-n(s) boolean AND S-1(n+1) and is of constant curvature (d{log vertical bar lambda(2)-(1-R)vertical bar(1/n)}/ds)(2) - lambda(2) + 1, where s is the arc length of an orthogonal trajectory of the family M-1(n-1) (s), E-n(s) is an n-dimensional linear subspace in R-1(n+2) which is parallel to a fixed E-n(s(0)) and u = vertical bar lambda(2) (1 - R)vertical bar(-1/n) satisfies the ordinary differental equation of order 2, d(2)u/ds(2) - u (+/- n-2/2 1/u(n) + R - 2) = 0.