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Hypersurfaces with isotropic Blaschke tensor. (English) Zbl 1242.53010

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 \(3\).

MSC:

53A30 Conformal differential geometry (MSC2010)
53B25 Local submanifolds

Citations:

Zbl 0912.53012
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Full Text: DOI

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
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