Guo, Zhen; Fang, Jianbo; Lin, Limiao Hypersurfaces with isotropic Blaschke tensor. (English) Zbl 1242.53010 J. Math. Soc. Japan 63, No. 4, 1155-1186 (2011). For a connected smooth \(n\)-dimensional hypersurface \(x: M^n \rightarrow \mathbb S^{n+1}\) without umbilical points, there exist three basic Möbius invariants, namely the Möbius metric \(g\), the Möbius form, the Blaschke tensor \(A\). We say that \(M\) is a hypersurface with isotropic Blaschke tensor if and only if there exists a function \(\lambda\) such that \(A= \lambda g\). Note that in this case this corresponds to the classical definition of an isotropic tensor in a Riemannian manifold, following O’Neill, saying that the length of \(A(v,\dots,v)\) is independent of the unit vector \(v\).The hypersurface is called constant isotropic if \(\lambda\) is a constant function on \(M\). In this case a classification was obtained in [C. Wang, Manuscr. Math. 96, No. 4, 517–534 (1998; Zbl 0912.53012)].Here the authors obtain a complete classification, up to Möbius equivalence, when \(\lambda\) is non constant, provided the dimension is at least \(3\). Reviewer: Luc Vrancken (Valenciennes) Cited in 8 Documents MSC: 53A30 Conformal differential geometry (MSC2010) 53B25 Local submanifolds Keywords:Möbius invariants; Möbius metric; Möbius form; Blaschke tensor; Möbius second fundamental form Citations:Zbl 0912.53012 PDF BibTeX XML Cite \textit{Z. Guo} et al., J. Math. Soc. Japan 63, No. 4, 1155--1186 (2011; Zbl 1242.53010) Full Text: DOI OpenURL References: [1] M. A. Akivis and V. V. Goldberg, Conformal differential geometry and its generalizations, Wiley, New York, 1996. · Zbl 0863.53002 [2] M. A. Akivis and V. V. Goldberg, A conformal differential invariant and the conformal rigidity of hypersurfaces, Proc. Amer. Math. Soc., 125 (1997), 2415-2424. · Zbl 0887.53030 [3] W. Blaschke, Vorlesungen über Differentialgeometrie, 3 , Springer-Verlag, Berlin, 1929. · JFM 55.0422.01 [4] B. Y. Chen, Total mean curvature and submanifolds of finite type, World Scientific, Singapore, 1984. · Zbl 0537.53049 [5] Z. Guo, H. Li and C. P. Wang, The Möbius characterizations of Willmore tori and Veronese submanifolds in the unit sphere, Pacific J. Math., 241 (2009), 227-242. · Zbl 1204.53011 [6] Z. Guo, H. Li and C. P. Wang, The second variation formula for Willmore submanifolds in \(S^{n}\), Results Math., 40 (2001), 205-225. · Zbl 1163.53312 [7] Z. J. Hu and H. Li, Submanifolds with constant Möbius scalar curvature in \(S^{n}\), Manuscripta Math., 111 (2003), 287-302. · Zbl 1041.53007 [8] Z. J. Hu and H. Li, Classification of hypersurfaces with parallel Möbius second fundamental form in \(S^{n+1}\), Sci. China Ser. A, 34 (2004), 28-39. · Zbl 1082.53016 [9] H. Li, H. L. Liu, C. P. Wang and G. S. Zhao, Möbius isoparametric hypersurfaces in \(S^{n+1}\) with two principal curvature, Acta Math. Sin. (Engl. Ser.), 18 (2002), 437-446. · Zbl 1030.53017 [10] H. Li and C. P. Wang, Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature, Manuscripta Math., 112 (2003), 1-13. · Zbl 1041.53008 [11] H. Li and C. P. Wang, Surfaces with vanishing Möbius form in \(S^n\), Acta Math. Sin. (Engl. Ser.), 19 (2003), 671-678. · Zbl 1078.53012 [12] H. Li, C. P. Wang and F. E. Wu, A Möbius characterization of Veronese surfaces in \(S^n\), Math. Ann., 319 (2001), 707-714. · Zbl 1031.53086 [13] H. L. Liu, C. P. Wang and G. S. Zhao, Möbius isotropic submanifolds in \(S^{n}\), Tohoku Math. J., 53 (2001), 553-569. · Zbl 1014.53010 [14] F. J. Pedit and T. J. Willmore, Conformal Geometry, Atti Sem. Mat. Fis. Univ. Modena, XXXVI (1988), 237-245. · Zbl 0665.53048 [15] C. P. Wang, Möbius geometry of submanifolds in \(S^{n}\), Manuscripta Math., 96 (1998), 517-534. · Zbl 0912.53012 [16] T. J. Willmore, Total curvature in Riemannian geometry, Ellis Horwood, Chichester, 1982. · Zbl 0501.53038 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.