×

Hypersurfaces of a Sasakian manifold – revisited. (English) Zbl 1504.53061

Summary: We study orientable hypersurfaces in a Sasakian manifold. The structure vector field \(\xi\) of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field \(\xi\) tangential to the hypersurface, and it also gives rise to a smooth function \(\sigma\) on the hypersurface, namely the projection of \(\xi\) on the unit normal vector field \(N\). Moreover, we have a second vector field tangent to the hypersurface, given by \(\mathbf{u}=-\varphi (N)\). In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere \(\mathbf{S}^{2n+1} \). Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature \(Ric ( \mathbf{u},\mathbf{u} )\) of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is \(\mathbf{S}^{2n+1} \).

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B25 Local submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adachi, T.; Kameda, M.; Maeda, S., Geometric meaning of Sasakian space form the view point of submanifold therory, Kodai Math. J., 33, 383-397 (2010) · Zbl 1213.53065 · doi:10.2996/kmj/1288962549
[2] Al-Solamy, F. R.; Khan, M. A., Semi-invariant warped product submanifolds of almost contact manifolds, J. Inequal. Appl., 2012 (2012) · Zbl 1276.53063 · doi:10.1186/1029-242X-2012-127
[3] Ali, A.; Alkhaldi, A. H.; Pişcoran, L.-I.; Ali, R., Eigenvalue inequalities for the p-Laplacian operator on C-totally real submanifolds in Sasakian space forms, Appl. Anal. (2020) · Zbl 1485.58009 · doi:10.1080/00036811.2020.1758307
[4] Ali, A.; Pişcoran, L.-I., Geometric classification of warped products isometrically immersed into Sasakian space forms, Math. Nachr., 292, 2, 234-251 (2019) · Zbl 1423.53068 · doi:10.1002/mana.201700121
[5] Alodan, H.; Chen, B.-Y.; Deshmukh, S.; Vîlcu, G.-E., A generalized Wintgen inequality for quaternionic CR-submanifolds, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 114, 3 (2020) · Zbl 1439.53054 · doi:10.1007/s13398-020-00866-8
[6] Alodan, H.; Deshmukh, S.; Turki, B. N.; Vilcu, G.-E., Hypersurfaces of a Sasakian manifold, Mathematics, 8 (2020) · doi:10.3390/math8060877
[7] Barbosa, E., On CMC free-boundary stable hypersurfaces in a Euclidean ball, Math. Ann., 372, 1-2, 179-187 (2018) · Zbl 1397.53013 · doi:10.1007/s00208-018-1658-z
[8] Bejancu, A.; Deshmukh, S., Real hypersurfaces of \(CP^n\) with non-negative Ricci curvature, Proc. Am. Math. Soc., 124, 1, 269-274 (1996) · Zbl 0866.53041 · doi:10.1090/S0002-9939-96-02886-9
[9] Bellettini, C.; Chodosh, O.; Wickramasekera, N., Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces, Adv. Math., 352, 133-157 (2019) · Zbl 1419.53056 · doi:10.1016/j.aim.2019.05.023
[10] Besse, A. L., Einstein Manifolds (1987), Berlin: Springer, Berlin · Zbl 0613.53001 · doi:10.1007/978-3-540-74311-8
[11] Bettiol, R.; Piccione, P.; Santoro, B., Deformations of free boundary CMC hypersurfaces, J. Geom. Anal., 27, 4, 3254-3284 (2017) · Zbl 1390.53056 · doi:10.1007/s12220-017-9804-5
[12] Biswas, I.; Schumacher, G., Vector bundles on Sasakian manifolds, Adv. Theor. Math. Phys., 14, 2, 541-561 (2010) · Zbl 1214.53037 · doi:10.4310/ATMP.2010.v14.n2.a5
[13] Blair, D. E., Riemannian Geometry of Contact and Symplectic Manifolds (2010), Boston: Birkhäuser, Boston · Zbl 1246.53001 · doi:10.1007/978-0-8176-4959-3
[14] Boyer, C.; Galicki, K., On Sasakian-Einstein geometry, Int. J. Math., 11, 7, 873-909 (2001) · Zbl 1022.53038 · doi:10.1142/S0129167X00000477
[15] Boyer, C.; Galicki, K., Sasakian Geometry (2008), Oxford: Oxford University Press, Oxford · Zbl 1155.53002
[16] Boyer, C.; Tonnesen-Friedman, C., The Sasaki join, Hamiltonian 2-forms, and constant scalar curvature, J. Geom. Anal., 26, 2, 1023-1060 (2016) · Zbl 1338.53068 · doi:10.1007/s12220-015-9583-9
[17] Chen, B.-Y., Pseudo-Riemannian Geometry, δ-Invariants and Applications (2011), Hackensack: World Scientific, Hackensack · Zbl 1245.53001 · doi:10.1142/8003
[18] Chen, B.-Y., Differential Geometry of Warped Product Manifolds and Submanifolds (2017), Hackensack: World Scientific, Hackensack · Zbl 1390.53001 · doi:10.1142/10419
[19] Collins, T. C.; Székelyhidi, G., Sasaki-Einstein metrics and K-stability, Geom. Topol., 23, 1339-1413 (2019) · Zbl 1432.32033 · doi:10.2140/gt.2019.23.1339
[20] de Almeida, S. C.; Brito, F. G.B.; Scherfner, M.; Weiss, S., On CMC hypersurfaces in \(S^{n+1}\) with constant Gauss-Kronecker curvature, Adv. Geom., 18, 2, 187-192 (2018) · Zbl 1391.53072 · doi:10.1515/advgeom-2017-0054
[21] Deshmukh, S., Real hypersurfaces in a Euclidean complex space form, Q. J. Math., 58, 303-307 (2007) · Zbl 1166.53312 · doi:10.1093/qmath/ham015
[22] Deshmukh, S., Real hypersurfaces of a complex space form, Monatshefte Math., 166, 93-106 (2012) · Zbl 1238.53008 · doi:10.1007/s00605-010-0269-x
[23] El Kacimi-Alaoui, A., Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compos. Math., 73, 1, 57-106 (1990) · Zbl 0697.57014
[24] Feng, K.; Zheng, T., Transverse fully nonlinear equations on Sasakian manifolds and applications, Adv. Math., 357 (2019) · Zbl 1429.53067 · doi:10.1016/j.aim.2019.106830
[25] Fetcu, D.; Oniciuc, C., Biharmonic hypersurfaces in Sasakian space forms, Differ. Geom. Appl., 27, 713-722 (2009) · Zbl 1188.53066 · doi:10.1016/j.difgeo.2009.03.011
[26] Futaki, A.; Ono, H.; Wang, G., Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Differ. Geom., 83, 3, 585-636 (2009) · Zbl 1188.53042 · doi:10.4310/jdg/1264601036
[27] García-Martínez, C.; Herrera, J., Rigidity and bifurcation results for CMC hypersurfaces in warped product spaces, J. Geom. Anal., 26, 2, 1186-1201 (2016) · Zbl 1339.53060 · doi:10.1007/s12220-015-9588-4
[28] Gervasio, C.; de Lima, E.; de Lima, H., Characterizations of complete CMC spacelike hypersurfaces satisfying an Okumura type inequality, Differ. Geom. Appl., 56, 295-307 (2018) · Zbl 1431.53061 · doi:10.1016/j.difgeo.2017.09.004
[29] Ghosh, A.; Sharma, R., Sasakian manifolds with purely transversal Bach tensor, J. Math. Phys., 58, 10 (2017) · Zbl 1381.53079 · doi:10.1063/1.4986492
[30] He, W.; Sun, S., Frankel conjecture and Sasaki geometry, Adv. Math., 291, 912-960 (2016) · Zbl 1333.53057 · doi:10.1016/j.aim.2015.11.053
[31] Kimura, M.; Maeda, S., On real hypersurface of a complex projective space, Math. Z., 202, 299-312 (1989) · Zbl 0661.53015 · doi:10.1007/BF01159962
[32] Lee, J. W.; Lee, C. W.; Vîlcu, G.-E., Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms, J. Geom. Phys., 155 (2020) · Zbl 1443.53008 · doi:10.1016/j.geomphys.2020.103768
[33] Maeda, S.; Tanabe, H.; Udagawa, S., Generating curves of minimal ruled real hypersurfaces in a nonflat complex space form, Can. Math. Bull., 62, 383-392 (2019) · Zbl 1414.53054 · doi:10.4153/CMB-2018-032-6
[34] Niebergall, R., Ryan, P.J.: Real hypersurfaces in complex space forms, Tight and Taut submanifolds (T.E. Cecil and S.S. Chern, eds.). Cambridge University Press, 233-305 (1998) (1998) · Zbl 0904.53005
[35] Obata, M., Conformal transformations of Riemannian manifolds, J. Differ. Geom., 4, 311-333 (1970) · Zbl 0205.52003 · doi:10.4310/jdg/1214429505
[36] Obata, M., The conjectures about conformal transformations, J. Differ. Geom., 6, 247-258 (1971) · Zbl 0236.53042 · doi:10.4310/jdg/1214430407
[37] Perdomo, O.; Tkachev, V., Algebraic CMC hypersurfaces of order 3 in Euclidean spaces, J. Geom., 110, 1 (2019) · Zbl 1475.53072 · doi:10.1007/s00022-018-0461-z
[38] Sasahara, T., Ricci curvature of real hypersurfaces in non-flat complex space forms, Mediterr. J. Math., 15 (2018) · Zbl 1394.53027 · doi:10.1007/s00009-018-1183-z
[39] Siddiqui, A. N.; Shahid, M. H.; Lee, J. W., Geometric inequalities for warped product bi-slant submanifolds with a warping function, J. Inequal. Appl., 2018 (2018) · Zbl 1498.53084 · doi:10.1186/s13660-018-1843-3
[40] Slesar, V.; Visinescu, M.; Vîlcu, G.-E., Toric data, Killing forms and complete integrability of geodesics in Sasaki-Einstein spaces \(Y^{p,q}\), Ann. Phys., 361, 548-562 (2015) · Zbl 1360.53050 · doi:10.1016/j.aop.2015.07.016
[41] Smoczyk, K.; Wang, G.; Zhang, Y., Sasaki-Ricci flow, Int. J. Math., 21, 7, 951-969 (2010) · Zbl 1209.53033 · doi:10.1142/S0129167X10006331
[42] Uddin, S.; Khan, K. A., An inequality for contact CR-warped product submanifolds of nearly cosymplectic manifolds, J. Inequal. Appl., 2012 (2012) · Zbl 1279.53053 · doi:10.1186/1029-242X-2012-304
[43] Uhlenbeck, K.; Yau, S.-T., On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math., 39, S1, S257-S293 (1986) · Zbl 0615.58045 · doi:10.1002/cpa.3160390714
[44] Wang, Y., Cyclic h-parallel shape and Ricci operators on real hypersurfaces in two-dimensional nonflat complex space forms, Pac. J. Math., 302, 335-352 (2019) · Zbl 1434.53021 · doi:10.2140/pjm.2019.302.335
[45] Watanabe, Y., Totally umbilical surfaces in normal contact Riemannian manifold, Kodai Math. Semin. Rep., 19, 474-487 (1967) · Zbl 0155.30801 · doi:10.2996/kmj/1138845504
[46] Yamaguchi, S., On hypersurfaces in Sasakian manifolds, Kodai Math. Semin. Rep., 21, 64-72 (1969) · Zbl 0184.47102 · doi:10.2996/kmj/1138845831
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.