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Laguerre geometry of hypersurfaces in $$mathbb{R}^{n}$$

Authors:	Tongzhu?Li,Changping?Wang author-information" > author-information__contact u-icon-before" > mailto:cpwang@math.pku.edu.cn" title=" cpwang@math.pku.edu.cn" itemprop=" email" data-track=" click" data-track-action=" Email author" data-track-label=" " >Email author

Affiliation:	(1) LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

Abstract:	Laguerre geometry of surfaces in $mathbb{R}^{3}$ 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 $mathbb{R}^{n}$ . For any umbilical free hypersurface $x: M rightarrow mathbb{R}^{n}$ with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in $mathbb{R}^{n}$ 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 $mathbb{R}^{n}$ , the semi-Euclidean space $mathbb{R}^{n}_{1}$ and the degenerate space $mathbb{R}^{n}_{0}$ we define three Laguerre space forms $Umathbb{R}^{n}$ , $Umathbb{R}^{n}_{1}$ and $Umathbb{R}^{n}_{0}$ and define the Laguerre embeddings $Umathbb{R}^{n}_{1} rightarrow U mathbb{R}^{n}$ and $Umathbb{R}^{n}_{0} rightarrow U mathbb{R}^{n}$ , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ, $mathbb{H}^{n}$ and $mathbb{R}^n$ (spaces of constant curvature) and conformal embeddings $mathbb{H}^n rightarrow S^n$ and $mathbb{R}^n rightarrow S^n$ [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 $mathbb{R}^n$ , $mathbb{R}^n_1$ and $mathbb{R}^n_0$ . As an example we show that minimal surfaces in $mathbb{R}^3_1$ or $mathbb{R}_0^3$ are Laguerre minimal in $mathbb{R}^3$ .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.

Keywords:	KeywordHeading" >Mathematics Subject Classification (2000) Primary 53A40 Secondary 53B25
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