SPACE-LIKE BLASCHKE ISOPARAMETRIC SUBMANIFOLDS IN THE LIGHT-CONE OF CONSTANT SCALAR CURVATURE

Let C-s(m+p+1 )subset of R-s+1(m+p+2) (m >= 2, p >= 1, 0 <= s <= p) be the standard (punched) light-cone in the Lorentzian space R-s+1(m+p+2) , and let Y : M-m -> C-s(m+p+1) be a space-like immersed submanifold of dimension m. Then, in addition to the induced metric g on M-m, there are three other important invariants of Y: the Blaschke tensor A, the conic second fundamental form B, and the conic Mobius form C; these are naturally defined by Y and are all invariant under the group of rigid motions on en C-s(m+p+1). In particular, g, A, B, C form a complete invariant system for Y, as was originally shown by C. P. Wang for the case in which s = 0. The submanifold Y is said to be Blaschke isoparametric if its conic Mobius form C vanishes identically and all of its Blaschke eigenvalues are constant. In this paper, we study the space-like Blaschke isoparametric submanifolds of a general codimension in the light-cone C(s)(m+p+1)for the extremal case in which s = p. We obtain a complete classification theorem for all the m-dimensional space-like Blaschke isoparametric submanifolds in C(s)(m+p+1 )of constant scalar curvature, and of two distinct Blaschke eigenvalues.