Browse > Article
http://dx.doi.org/10.4134/BKMS.b180491

A NOTE ON COMPACT MÖBIUS HOMOGENEOUS SUBMANIFOLDS IN 𝕊n+1  

Ji, Xiu (Department of Mathematics Beijing Institute of Technology)
Li, TongZhu (Department of Mathematics Beijing Institute of Technology)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.3, 2019 , pp. 681-689 More about this Journal
Abstract
The $M{\ddot{o}}bius$ homogeneous submanifold in ${\mathbb{S}}^{n+1}$ is an orbit of a subgroup of the $M{\ddot{o}}bius$ transformation group of ${\mathbb{S}}^{n+1}$. In this note, We prove that a compact $M{\ddot{o}}bius$ homogeneous submanifold in ${\mathbb{S}}^{n+1}$ is the image of a $M{\ddot{o}}bius$ transformation of the isometric homogeneous submanifold in ${\mathbb{S}}^{n+1}$.
Keywords
$M{\ddot{o}}bius$ transformation group; isometric transformation group; $M{\ddot{o}}bius$ homogeneous hypersurfaces; homogeneous hypersurfaces;
Citations & Related Records
연도 인용수 순위
  • Reference
1 T. E. Cecil, Lie Sphere Geometry, Universitext, Springer-Verlag, New York, 1992.
2 U. Hertrich-Jeromin, Introduction to Mobius Differential Geometry, London Mathematical Society Lecture Note Series, 300, Cambridge University Press, Cambridge, 2003.
3 W. Hsiang and H. B. Lawson, Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry 5 (1971), 1-38.   DOI
4 T. Li, Mobius homogeneous hypersurfaces with three distinct principal curvatures in $\mathbb{S}^{n+1}$, Chin. Ann. Math. Ser. B 38 (2017), no. 5, 1131-1144.   DOI
5 T. Li, X. Ma, and C. Wang, Mobius homogeneous hypersurfaces with two distinct principal curvatures in $S^{n+1}$, Ark. Mat. 51 (2013), no. 2, 315-328.   DOI
6 T. Li and C. Wang, Classification of Mobius homogeneous hypersurfaces in a 5-dimensional sphere, Houston J. Math. 40 (2014), no. 4, 1127-1146.
7 X. Ma, F. Pedit, and P. Wang, Mobius homogeneous Willmore 2-spheres, Bull. London Math. Soc., 2018; doi:10.1112/blms.12155.   DOI
8 H. F. Munzner, Isoparametrische Hyper achen in Spharen, Math. Ann. 251 (1980), no. 1, 57-71.   DOI
9 B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.
10 R. Sulanke, Mobius geometry. V. Homogeneous surfaces in the Mobius space $S^3$, in Topics in differential geometry, Vol. I, II (Debrecen, 1984), 1141-1154, Colloq. Math. Soc. Janos Bolyai, 46, North-Holland, Amsterdam, 1988.
11 R. Takagi and T. Takahashi, On the principal curvatures of homogeneous hypersurfaces in a sphere, in Differential geometry (in honor of Kentaro Yano), 469-481, Kinokuniya, Tokyo, 1972.
12 C.Wang, Moebius geometry of submanifolds in $S^n$, Manuscripta Math. 96 (1998), no. 4, 517-534.   DOI