Laguerre geometry of hypersurfaces in ℝn

Tongzhu Li, Changping Wang*

*此作品的通讯作者

科研成果: 期刊稿件文章同行评审

30 引用 (Scopus)

摘要

Laguerre geometry of surfaces in ℝ3 is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321-4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123-138 (1999); Int. J. Math. 11(7), 911-924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281-293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in ℝn. For any umbilical free hypersurface x: M → ℝn/ with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator double script S sign : TM → TM, and show that g doubl script S sign is a complete Laguerre invariant system for hypersurfaces in ℝn with n ≥ 4. We calculate the Euler-Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space ℝn, the semi-Euclidean space ℝ1n and the degenerate space ℝ0 n we define three Laguerre space forms Uℝn , Uℝ1n and ℝ0nn and define the Laguerre embeddings Uℝ1n → U ℝn and ℝ1n, analogously to what happens in the Moebius geometry where we have Moebius space forms S n , ℍn and ℝn (spaces of constant curvature) and conformal embeddings ℍn → Sn and ℝn → Sn [cf. Liu et al. in Tohoku Math. J. 53, 553-569 (2001) and Wang in Manuscr. Math. 96, 517-534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in ℝn, ℝ1n and ℝ0 n. As an example we show that minimal surfaces in ℝ13 or ℝ03 are Laguerre minimal in ℝ3.

源语言英语
页(从-至)73-95
页数23
期刊Manuscripta Mathematica
122
1
DOI
出版状态已出版 - 1月 2007
已对外发布

指纹

探究 'Laguerre geometry of hypersurfaces in ℝn' 的科研主题。它们共同构成独一无二的指纹。

引用此

Li, T., & Wang, C. (2007). Laguerre geometry of hypersurfaces in ℝn. Manuscripta Mathematica, 122(1), 73-95. https://doi.org/10.1007/s00229-006-0058-y