Abstract
Let \(\mathfrak{C}_s^{m + p + 1} \subset \mathbb{R}_{s + 1}^{m + p + 2}(m \ge 2,\,p \ge 1,\,0 \le s \le p)\) be the standard (punched) light-cone in the Lorentzian space \(\mathbb{R}_{s + 1}^{m + p + 2}\), and let \(Y:{M^m} \to \mathfrak{C}_s^{m + p + 1}\) be a space-like immersed submanifold of dimension m. Then, in addition to the induced metric g on Mm, 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 \(\mathfrak{C}_s^{m + p + 1}\). 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 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 \(\mathfrak{C}_s^{m + p + 1}\) 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 \(\mathfrak{C}_p^{m + p + 1}\) of constant scalar curvature, and of two distinct Blaschke eigenvalues.
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Research supported by Foundation of Natural Sciences of China (11671121, 11871197 and 11431009).
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Song, H., Liu, X. Space-Like Blaschke Isoparametric Submanifolds in the Light-Cone of Constant Scalar Curvature. Acta Math Sci 42, 1547–1568 (2022). https://doi.org/10.1007/s10473-022-0415-2
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DOI: https://doi.org/10.1007/s10473-022-0415-2
Key words
- Conic Möbius form
- parallel Blaschke tensor
- induced metric
- conic second fundamental form
- stationary submanifolds
- constant scalar curvature