Time-Like Conformal Homogeneous Hypersurfaces with Three Distinct Principal Curvatures
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Time-Like Conformal Homogeneous Hypersurfaces with Three Distinct Principal Curvatures∗ Yanbin LIN1
¨2 Ying LU
Changping WANG3
Abstract A hypersurface x(M ) in Lorentzian space R14 is called conformal homogeneous, if for any two points p, q on M , there exists σ, a conformal transformation of R14 , such that σ(x(M )) = x(M ), σ(x(p)) = x(q). In this paper, the authors give a complete classification for regular time-like conformal homogeneous hypersurfaces in R14 with three distinct principal curvatures. Keywords Lorentzian metric, Conformal metric, Conformal space form, Conformal homogeneous, Time-like hypersurface 2000 MR Subject Classification 53A30, 53B25
1 Introduction as
Let {R26 , h·, ·i} be a Lorentzian space form of dimension 6 and the inner product is defined hu, vi = u1 v1 + u2 v2 + u3 v3 + u4 v4 − u5 v5 − u6 v6 .
The conformal space Q41 is defined in the light cone by Q41 = {[u] ∈ RP 5 | u ∈ R26 , hu, ui = 0}, which is the conformal compactification of Lorentzian space forms R14 , S14 and H14 . The conformal transformation group is therefore isomorphic to O(4, 2)/{±1}. Since the hypersurfaces in three Lorentzian space forms are conformally equivalent to each other, we choose R14 as the ambient space to study the conformal properties of hypersurfaces. More details on the conformal space Qm 1 can be found in [3, 6]. Suppose that x : M 3 → (R14 , h , i1 ) is a time-like hypersurface in Lorentzian space form, in which h , i1 is a Lorentzian inner product with signatures (+, +, +, −). If at every point p, {ei } is a basis of Tp M with dual basis {ω i } and n is the space-like unit normal vector, then there is Manuscript received July 11, 2018. of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, Fujian, China. E-mail: [email protected] 2 School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, China. E-mail: [email protected] 3 College of Mathematics and Informatics, Fujian Normal University, Fuzhou 350117, China. E-mail: [email protected] ∗ The first author is supported by the Principal’s Fund (No. KJ2020002), the second is supported by the National Natural Science Foundation of China (Nos. 11671330 and 11871405), the third is supported by the National Natural Science Foundation of China (Nos. 11831005, 1196131001). 1 School
680
Y. B. Lin, Y. L¨ u and C. P. Wang
a naturally induced Lorentzian metric on M , g = hdx, dxi1 =
X
gij ω i ⊗ ω j .
i,j
The structure equations are in form of X X j dx = ω i ei (x), dei (x) = ωi ej (x) + hij ω j n, i
in which h =
P i,j
j
X
dn = −
hij ω i ⊗ ω j is the second fundamental form and S =
Sij ω i ej (x),
i,j
P i,j
Sij ω i ⊗ ej is the shape
operator. According to the algebraic lemma in [9], we have the following lemma. Lemma 1.1 There exists a basis {ei } such that the matrices of shape operator and induced metric are exactly in one of the following forms: λ1 −1 , g = λ2 1 , λ2 ≤ λ3 ; (i)
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