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SPACE-LIKE BLASCHKE ISOPARAMETRIC SUBMANIFOLDS IN THE LIGHT-CONE OF CONSTANT SCALAR CURVATURE
宋虹儒, 刘西民
数学物理学报(英文版). 2022 (4):
1547-1568.
DOI: 10.1007/s10473-022-0415-2
Let \mathfrak{C}^{m+p+1}_s\subset\mathbb{R}^{m+p+2}_{s+1} (m\geq 2, p\geq 1, 0\leq s\leq p) be the standard (punched) light-cone in the Lorentzian space \mathbb{R}^{m+p+2}_{s+1}, and let Y:M^m\to \mathfrak{C}^{m+p+1}_s 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 Möbius form C; these are naturally defined by Y and are all invariant under the group of rigid motions on \mathfrak{C}^{m+p+1}_s. 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 Möbius 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 \mathfrak{C}^{m+p+1}_s 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 \mathfrak{C}^{m+p+1}_p of constant scalar curvature, and of two distinct Blaschke eigenvalues.
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