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A Möbius characterization of submanifolds in real space forms with parallel mean curvature and constant scalar curvature. (English) Zbl 1088.53008
The authors provide a Möbius geometric characterization of submanifolds with parallel mean curvature vector field and constant scalar curvature in space forms in terms of the Möbius invariants for submanifolds introduced by C. P. Wang [Manuscr. Math. 96, 517–534 (1998; Zbl 0912.53012)]. Note that the space form geometries naturally appear as subgeometries of Möbius geometry.
In this way the authors generalize earlier theorems by H. L. Liu, C. P. Wang and G. S. Zhao [Tohoku Math. J. 53, 553–569 (2001; Zbl 1014.53010)] and by H. Li and C. P. Wang [Manuscr. Math. 112, 1–13 (2003; Zbl 1041.53008)].

MSC:
53B25 Local submanifolds
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[1] Akivis, M.A., Goldberg, V.V.: Conformal differential geometry and its generalizations. Wiley, New York, 1996 · Zbl 0863.53002
[2] Akivis, M.A., Goldberg, V.V.: A conformal differential invariant and the conformal rigidity of hypersurfaces. Proc. Am. Math. Soc. 125, 2415–2424 (1997) · Zbl 0887.53030 · doi:10.1090/S0002-9939-97-03828-8
[3] Blaschke, W.: Vorlesungen über Differentialgeometrie. Vol. 3, Springer Berlin, 1929 · JFM 55.0422.01
[4] Chen, B.Y.: Total Mean Curvature and Submanifolds of Finite Type. World Scientific, Singapore, 1984 · Zbl 0537.53049
[5] Guo, Z. Li, H., Wang, C.P.: The second variation formula for Willmore submanifolds in Sn. Results Math. 40, 205–225 (2001) · Zbl 1163.53312
[6] Hu, Z.J., Li, H.: Submanifolds with constant Möbius scalar curvature in Sn. Manuscripta Math. 111, 287–302 (2003) · Zbl 1041.53007
[7] Kulkarni, R.S., Pinkall, U.: Conformal Geometry, Aspects Math. E12, Fredr. Vieweg & Son, Braunschweig, 1988
[8] Li, H.: Hypersurfaces with constant scalar curvature in spaces forms. Math. Ann. 305, 665–672 (1996) · Zbl 0864.53040 · doi:10.1007/BF01444220
[9] Li, H.: Willmore hypesurfaces in a sphere. Asian J. Math. 5, 365–377 (2001) · Zbl 1025.53031
[10] Li, H.: Willmore submanifolds in a sphere. Math. Res. Lett. 9, 771–790 (2002) · Zbl 1056.53040
[11] Li, H., Liu, H.L., Wang, C.P., Zhao, G.S.: Möbius isoparametric hypersurfaces in Sn+1 with two distinct principal curvatures. Acta Math. Sinica, English Series, 18, 437–446 (2002) · Zbl 1030.53017 · doi:10.1007/s10114-002-0173-y
[12] Li, H., Wang, C.P.: Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature. Manuscripta Math. 112, 1–13 (2003) · Zbl 1041.53008 · doi:10.1007/s00229-003-0383-3
[13] Li, H., Wang, C.P.: Surfaces with vanishing Möbius form in Sn. Acta Math. Sinica, English Series 19, 671–678 (2003) · Zbl 1078.53012 · doi:10.1007/s10114-003-0309-8
[14] Li, H., Wang, C.P., Wu, F.E.: A Moebius characterization of Veronese surfaces in Sn. Math. Ann. 319, 707–714 (2001) · Zbl 1031.53086 · doi:10.1007/PL00004455
[15] Liu, H.L., Wang, C.P., Zhao, G.S.: Möbius isotropic submanifolds in Sn. Tohoku Math. J. 53, 553–569 (2001) · Zbl 1014.53010 · doi:10.2748/tmj/1113247800
[16] Wang, C.P.: Möbius geometry of submanifolds in Sn. Manuscripta Math. 96, 517–534 (1998) · Zbl 0912.53012 · doi:10.1007/s002290050080
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