Abstract
Let \(x: M \rightarrow \mathbb {R}^n\) be an \((n-1)\)-dimensional umbilic free hypersurface with non-zero principal curvatures in \(\mathbb {R}^n\), \(\mathbf B\) be the Laguerre second fundamental form, \(\mathbf L\) be the Laguerre tensor and \({\mathbf D}={\mathbf L}+\lambda {\mathbf B}\) be the para-Laguerre tensor of the immersion x, where \(\lambda \) is a constant. In this paper, we study the Laguerre isoparametric hypersurfaces, constant para-Laguerre eigenvalues hypersurfaces and Dupin hypersurfaces in \(\mathbb {R}^n\) and obtain some classification theorems.
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References
Blaschke, W.: Vorlesungenüber Differential Geometrie, vol. 3. Springer, Berlin (1929)
Bobenko, A.I., Pottmann, H., Wallner, J.: A curvature theory for discrete surfaces based on mesh parallelity. Math. Ann. 348(1), 1–24 (2010)
Li, T.Z., Li, H., Wang, C.P.: A note on Blaschke isoparametric hypersurfaces. Int. J. Math. 25(12), 1–9 (2014)
Li, T.Z., Wang, C.P.: Laguerre geometry of hypersurfaces in \(\mathbb{R}^n\). Manuscr. Math. 122, 73–95 (2007)
Li, T.Z., Li, H., Wang, C.P.: Classification of hypersurfaces with parallel Laguerre second fundamental form in \(\mathbb{R}^n\). Differ. Geom. Appl. 28, 148–157 (2010)
Li, T.Z., Li, H., Wang, C.P.: Classification of hypersurfaces with constant Laguerre eigenvalues in \(\mathbb{R}^n\). Sci. China Math. 54, 1129–1144 (2011)
Musso, E., Nicolodi, L.: A variational problem for surfaces in Laguerre geometry. Trans. Am. Math. soc. 348, 4321–4337 (1996)
Musso, E., Nicolodi, L.: Laguerre geometry of surfaces with plane lines of curvature. Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999)
Musso, E., Nicolodi, L.: Deformation and applicability of surfaces in Lie sphere geometry. Tohoku Math. J. (2) 58(2), 161–187 (2006)
Musso, E., Nicolodi, L.: Holomorphic differentials and Laguerre deformation of surfaces. Math. Z. 284(3–4), 1089–1110 (2016)
Pottmann, H., Grohs, P., Blaschitz, B.: Edge offset meshes in Laguerre geometry. Adv. Comput. Math. 33(1), 45–73 (2010)
Rogers, C., Szereszewski, A.: A Bäcklund transformation for L-isothermic surfaces. J. Phys. A 42(40), 404015 (2009)
Shu, S.C.: Hypersurfaces with parallel para-Laguerre tensor in \(\mathbb{R}^n\). Math. Nachr. 286, 17–18 (2013)
Song, Y.P.: Laguerre isoparametric hypersurfaces in \(\mathbb{R}^n\) with two distinct non-zero principal curvatures. Acta Math. Sin. Engl. Ser. 30, 169–180 (2014)
Szereszewski, A.: L-isothermic and L-minimal surfaces. J. Phys. A 42(11), 115203 (2009)
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The author would like to thank the referee for his / her many valuable comments and suggestions that really improve the paper.
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Project supported by NSF of Shaanxi Province (SJ08A31).
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Shu, S. Laguerre isoparametric and Dupin hypersurfaces in \(\mathbb {R}^n\) . RACSAM 112, 361–372 (2018). https://doi.org/10.1007/s13398-017-0386-7
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DOI: https://doi.org/10.1007/s13398-017-0386-7