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.
Hongru SONG
,
Ximin LIU
. SPACE-LIKE BLASCHKE ISOPARAMETRIC SUBMANIFOLDS IN THE LIGHT-CONE OF CONSTANT SCALAR CURVATURE[J]. Acta mathematica scientia, Series B, 2022
, 42(4)
: 1547
-1568
.
DOI: 10.1007/s10473-022-0415-2
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