Totally umbilical hypersurfaces of \(\mathrm{Spin}^c\) manifolds carrying special spinor fields. (English) Zbl 1461.53037
In the paper under review, the authors consider totally umbilical compact hypersurfaces \((M^m,g)\) of Spin\(^c\) manifolds carrying \(\alpha\)-Killing spinors (\(\alpha\) is real number or purely imaginary). They show that such hypersurfaces have constant mean curvature, if \(m\geq 4\) (for \(\alpha\) real) and \(m\geq 2\) (for \(\alpha\) purely imaginary). As a consequence, they deduce that every connected totally umbilical hypersurface of a Kähler or a Sasakian manifold have constant mean curvature. As another application, they give some non-existence results of extrinsic hyperspheres in some complete manifolds.
Reviewer: Georges Habib (Beirut)
MSC:
| 53C27 | Spin and Spin\({}^c\) geometry |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
Keywords:
totally umbilical hypersurfaces; constant mean curvature; \(\mathrm{Spin}^c\) manifolds with special spinor fields; differential forms; extrinsic hypersphereReferences:
| [1] | Bär, C., Real Killing spinors and holonomy, Comm. Math. Phys.154(3) (1993) 509-521. · Zbl 0778.53037 |
| [2] | Bär, C., Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Glob. Anal. Geom.16 (1998) 573-596. · Zbl 0921.58065 |
| [3] | Baum, H., Complete Riemannian manifolds with imaginary Killing spinors, Ann. Global Anal. Geom.7(3) (1989) 205-226. · Zbl 0694.53043 |
| [4] | Baum, H., Odd-dimensional Riemannian manifolds with imaginary Killing spinors, Ann. Global Anal. Geom.7(2) (1989) 141-153. · Zbl 0708.53039 |
| [5] | Baum, H., Variétés riemanniennes admettant des spineurs de Killing imaginaires, C. R. Acad. Sci. Paris Sér. I Math.309(1) (1989) 47-49. · Zbl 0676.53053 |
| [6] | Baum, H., Friedrich, T., Grunewald, R. and Kath, I., Twistor and Killing Spinors on Riemannian Manifolds, , Vol. 108 (Humboldt Universität Sektion Mathematik, Berlin, 1990). · Zbl 0705.53004 |
| [7] | Berndt, J., Console, S. and Olmos, C., Submanifolds and Holonomy, Vol. 434 (Chapman and Hall, Boca Raton, 2003). · Zbl 1043.53044 |
| [8] | Bertola, M. and Gouthier, D., Warped products with special Riemannian curvature, Bol. Soc. Bras. Mat.32(1) (2001) 45-62. · Zbl 1107.53025 |
| [9] | J. P. Bourguignon, O. Hijazi, J. L. Milhorat and A. Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry, EMS Monographs in Mathematics (2015). · Zbl 1348.53001 |
| [10] | Boyer, C. and Galicki, K., 3-Sasakian manifolds, Surv. Diff. Geom.7 (1999) 123. · Zbl 1008.53047 |
| [11] | Chen, B. Y., Extrinsic spheres in compact symmetric spaces are intrinsic spheres, Michigan Math. J.24(3) (1977) 265-271. · Zbl 0389.53025 |
| [12] | Chen, B. Y., Odd-dimensional extrinsic spheres in Kähler manifolds, Rend. Mat. (6)12(2) (1979) 201-207. · Zbl 0447.53052 |
| [13] | Chen, B. Y., Classification of totally umbilical submanifolds in symmetric spaces, J. Austral. Math. Soc. (Ser. A)30 (1980) 129-136. · Zbl 0459.53039 |
| [14] | Chen, B. Y., Totally umbilical submanifolds of Kähler manifolds, Arch. Math.36(1) (1981) 83-91. · Zbl 0459.53044 |
| [15] | Chen, B. Y. and Nagano, T., Totally geodesic submanifolds of symmetric spaces, II, Duke Math. J.45(2) (1978) 405-425. · Zbl 0384.53024 |
| [16] | Friedrich, T., Der erste Eigenwert des Dirac-operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nach.97 (1980) 117-146. · Zbl 0462.53027 |
| [17] | Friedrich, T., Dirac Operators in Riemannian Geometry, , Vol. 25 (AMS, Providence, RI, 2000), xvi+195 pp. · Zbl 0949.58032 |
| [18] | N. Ginoux, Opérateurs de Dirac sur les sous-variétés, Ph. D thesis, Institut Élie Cartan, Nancy (2002). |
| [19] | Große, N. and Nakad, R., Complex generalized Killing spinors on Riemannian Spin^c manifolds, Results Math.67(1) (2015) 177-195. · Zbl 1318.53046 |
| [20] | Herzlich, M. and Moroianu, A., Generalized Killing spinors and conformal eigenvalue estimates for Spin^c manifolds, Ann. Glob. Anal. Geom.17 (1999) 341-370. · Zbl 0988.53020 |
| [21] | O. Hijazi, Opérateurs de Dirac sur les variétés riemanniennes: minoration des valeurs propres, Ph. D thesis, Ecole Polytechnique (1984). |
| [22] | Hijazi, O., A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Commun. Math. Phys.104 (1986) 151-162. · Zbl 0593.58040 |
| [23] | Hijazi, O. and Montiel, S., A spinorial characterization of hyperspheres, Calc. Var. Partial Differential Equation.48 (2013) 527-544. · Zbl 1286.53057 |
| [24] | Hijazi, O. and Montiel, S., A holographic principle for the existence of parallel spinors and an inequality of Shi-Tam type, Asian J. Math.18(3) (2014) 489-506. · Zbl 1315.53050 |
| [25] | Hijazi, O., Montiel, S. and Roldan, A., Dirac operators on hypersurfaces of manifolds with negative scalar curvature, Ann. Global Anal. Geom.23 (2003) 247-264. · Zbl 1032.53040 |
| [26] | Hijazi, O., Montiel, S. and Urbano, F., Spin^c geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds, Math. Z.253(4) (2006) 821-853. · Zbl 1101.53029 |
| [27] | Hijazi, O., Montiel, S. and Zhang, X., Dirac operator on embedded hypersurfaces, Math. Res. Lett.8 (2001) 195-208. · Zbl 0988.53019 |
| [28] | Jentsch, T., Moroianu, A. and Semmelmann, U., Extrinsic hypersphere in manifolds with special holonomy, Differ. Geom. Appl.31 (2013) 104-111. · Zbl 1275.53042 |
| [29] | Koiso, N., Hypersurfaces of Einstein manifolds, Ann. Sci. École Norm. Sup.14(4) (1981) 433-443. · Zbl 0493.53041 |
| [30] | D. Kowalczyk, Submanifolds of product spaces, Ph.D. thesis, Katolieke Universiteit LEUVEN (2011). |
| [31] | Kowalski, O., Properties of hypersurfaces which are characteristic for spaces of constant curvature, Ann. Scuola Norm. Superiore Pisa, Classe di Scienze, Serie (3)26(1) (1972) 233-245. · Zbl 0233.53009 |
| [32] | Kühnel, W. and Rademacher, H.-B, Twistor Spinors with zeros, Int. J. Math.5 (1994) 877-895. · Zbl 0818.53054 |
| [33] | Lawson, H. B. and Michelson, M. L., Spin Geometry (Princeton University Press, Princeton, New Jersey, 1989). · Zbl 0688.57001 |
| [34] | Lichnerowicz, A., Sur les zéros des spineurs-twisteurs, C.R. Acad. Sci. Paris310 (1990) 19-22. · Zbl 0687.53043 |
| [35] | Moroianu, A., Parallel and Killing spinors on Spin^c manifolds, Comm. Math. Phys.187(2) (1997) 417-427. · Zbl 0888.53035 |
| [36] | Nakad, R., The energy-momentum tensor on Spin^c manifolds, Int. J. Geom. Methods Mod. Phys.8(2) (2011) 345-365. · Zbl 1221.53082 |
| [37] | R. Nakad, Sous-variétés spéciales des variétés spinorielles complexes, Ph.D. thesis, Université Henri Poincaré-Nancy I, (2011). |
| [38] | Okumuara, M., Totally umbilical hypersurfaces of a locally product Riemannian manifolds, Kodai Math. Sem. Rep.19 (1967) 35-42. · Zbl 0145.41902 |
| [39] | O’Neill, B., Semi-Riemannian Geometry (Academic Press, New York, 1983). · Zbl 0531.53051 |
| [40] | Rademacher, H. B., Generalized Killing spinors with imaginary Killing function and conformal Killing fields, in Global Differential Geometry and Global Analysis, , Vol. 1481 (Springer, Berlin, 1991), pp. 192-198. · Zbl 0762.53024 |
| [41] | Roth, J. and Nakad, R., Hypersurfaces of Spin^c manifolds and Lawson type correspondence, Ann. Glob. Anal. Geom.42(3) (2012) 421-442. · Zbl 1258.53048 |
| [42] | Souam, R. and Toubiana, E., Totally umbilic surfaces in homogeneous 3-manifolds, Comment. Math. Helv.84 (2009) 673-704. · Zbl 1170.53030 |
| [43] | Tojo, K., Extrinsic hyperspheres of naturally reductive homogeneous spaces, Tokyo J. Math.20(1) (1997) 35-43. · Zbl 0881.53050 |
| [44] | Tsukada, K., Totally geodesic hypersurfaces of naturally reductive homogeneous spaces, Osaka J. Math.33(3) (1996) 697-707. · Zbl 0871.53042 |
| [45] | Yamaguchi, S., Nemoto, H. and Kawabata, N., Extrinsic spheres in a Kähler manifold, Michigan Math. J.31(1) (1984) 15-19. · Zbl 0578.53038 |
| [46] | Wang, M., Parallel spinors and parallel forms, Ann. Glob. Anal. Geom.7 (1989) 59-68. · Zbl 0688.53007 |
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