×

Ricci curvature on warped product submanifolds in spheres with geometric applications. (English) Zbl 1427.53067

Summary: The goal of this paper is to construct a fundamental theorem for the Ricci curvature inequality via partially minimal isometric warped product immersions into an \(m\)-dimensional unit sphere \(\mathbb{S}^m\), involving the Laplacian of a well defined warping function, the squared norm of a warping function and the squared norm of the mean curvature. Moreover, the equality cases are discussed in detail and some applications are also derived due to involvement of the warping function. As applications, we provide sufficient condition that the base \(N_1^p\) is isometric to the sphere \(\mathbb{S}^p(\frac{\lambda_1}{p})\) with constant sectional curvature \(c = \frac{\lambda_1}{p}\). The obtained results in the paper give the partial solution of Ricci curvature conjecture, also known as Chen-Ricci inequality obtained by B.-Y. Chen [Glasg. Math. J. 41, No. 1, 33–41 (1999; Zbl 0962.53015)].

MSC:

53C40 Global submanifolds
53C20 Global Riemannian geometry, including pinching
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B25 Local submanifolds

Citations:

Zbl 0962.53015
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexander, S.; Maltz, R., Isometric immersions of Riemannian products in Euclidean space, J. Differential Geom., 11, 47-57 (1976) · Zbl 0334.53053
[2] Ali, A.; Laurian-Ioan, P., Geometry of warped product immersions of Kenmotsu space forms and its applications to slant immersions, J. Geom. Phy., 114, 276-290 (2017) · Zbl 1359.53043
[3] Ali, A.; Laurian-Ioan, P., Geometric classification of warped products isometrically immersed in Sasakian space forms, Math. Nachr., 292, 234-251 (2018) · Zbl 1423.53068
[4] Ali, A.; Lee, J. W.; Alkhaldi, A. H., Geometric classification of warped product submanifolds of nearly Kaehler manifolds with a slant fiber, Int. J. Geom. Methods Mod. Phys., 16, 02, Article 1950031 pp. (2019) · Zbl 1408.53072
[5] Ali, A.; Othman, W. A.M.; Ozel, C., Some inequalities for warped product pseudo-slant submanifolds of nearly Kenmotsu manifolds, J. Inequal. Appl., 2015, 291 (2015) · Zbl 1335.53022
[6] Ali, A.; Ozel, C., Geometry of warped product pointwise semi-slant submanifolds of cosymplectic manifolds and its applications, Int. J. Geom. Methods Mod. Phys., 14, 03, Article 1750042 pp. (2017) · Zbl 1362.53062
[7] Ali, A.; Uddin, S.; Othman, W. A.M., Geometry of warped product pointwise semi-slant submanifolds of Kaehler manifolds, Filomat, 32, 3771-3788 (2017) · Zbl 1488.53176
[8] Aquib, M.; Lee, J. W.; Vilcu, G. E.; Yoon, W., Classification of Casorati ideal Lagrangian submanifolds in complex space forms, Differential Geom. Appl., 63, 30-49 (2019) · Zbl 1419.53044
[9] Berger, M., Les variétés riemanniennes \((\frac{1}{4})\)-pincées, Ann. Sc. Norm. Super. Pisa Cl. Sci., 14, 4, 161-170 (1960) · Zbl 0096.15502
[10] Bishop, R. L.; O’Neil, B., Manifolds of negative curvature, Trans. Amer. Math. Soc., 145, 1-9 (1969) · Zbl 0191.52002
[11] Calin, O.; Chang, D. C., Geometric Mechanics on Riemannian Manifolds: Applications to Partial Differential Equations (2006), Springer Science & Business Media
[12] Cheeger, J.; Coding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Maths, 144, 189-237 (1996) · Zbl 0865.53037
[13] Chen, B. Y., A general inequality for submanifolds in complex space forms and its applications, Arch. Math., 67, 519-528 (1996) · Zbl 0871.53043
[14] Chen, B. Y., Mean curvature and shape operator of isometric immersions in real space forms, Glasg. Math. J., 38, 87-97 (1996) · Zbl 0866.53038
[15] Chen, B. Y., Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasg. Math. J., 41, 33-41 (1999) · Zbl 0962.53015
[16] Chen, B. Y., Geometry of warped product CR-submanifolds in Kaehler manifolds, Monatsh. Math., 133, 177-195 (2001) · Zbl 0996.53044
[17] Chen, B. Y., On isometric minimal immersions from warped products into real space forms, Proc. Edinb. Math. Soc., 45, 03, 579-587 (2002) · Zbl 1022.53022
[18] Chen, B. Y., Differential Geometry of Warped Product Manifolds and Submanifolds (2017), World Scientific · Zbl 1390.53001
[19] B.Y. Chen, F. Dillen, L. Verstraelen, L. Vrancken, Characterization of Riemannian space forms, Einstein spaces and conformally flate spaces. Proc. Amer. Math. Soc. 128 (2) (199) 589-598.; B.Y. Chen, F. Dillen, L. Verstraelen, L. Vrancken, Characterization of Riemannian space forms, Einstein spaces and conformally flate spaces. Proc. Amer. Math. Soc. 128 (2) (199) 589-598. · Zbl 1007.53019
[20] Chen, B. Y.; geometry, Pseudo-Riemannian., Pseudo-Riemannian Geometry, \( \delta \)-Invariants and Applications (2011), World Scientific: World Scientific Hackensack, NJ · Zbl 1245.53001
[21] Cheng, S. S., Pectrum of the Laplacian and its Applications to Differential Geometry (1974), Univ. of California: Univ. of California Berkeley, (Ph.D. Dissertation)
[22] Cheng, S. Y., Eigenvalue comparison theorem and its geometric applications, Math. Z., 143, 289-297 (1975) · Zbl 0329.53035
[23] Crasmareanu, M., Last multipliers for Riemannian geometries, Dirichlet forms and Markov diffusion semigroups, J. Geom. Anal., 27, 2618 (2017) · Zbl 1390.53026
[24] Dajczer, M.; Tojeiro, R., Isometric immersions in codimension two of warped products into space forms, Illinois J. Math., 48, 711-746 (2004) · Zbl 1067.53008
[25] Dajczer, M.; Vlachos, Th., Isometric immersions of warped products, Proc. Amer. Math. Soc., 141, 1795-1803 (2013) · Zbl 1262.53047
[26] Garcia-Rio, E.; Kupeli, D. N.; Unal, B., On a differential equation characterizing Euclidean sphere, J. Differential Equations, 194, 287-299 (2003) · Zbl 1058.53027
[27] Hamilton, R. H., Three-manifolds with positive Ricci curvature, J. Differential Geom., 17, 255-306 (1982) · Zbl 0504.53034
[28] Kim, D. S.; Kim, Y. H., Compact Einstein warped product spaces with nonpositive scalar curvature, Proc. Amer. Math. Soc., 131, 2573-2576 (2003) · Zbl 1029.53027
[29] Leung, P. F., On a relation between the topology and the instrinsic and extrinsic geometries of compact submanifold, Proc. Endinb. Math. Soc., 28, 305-311 (1985) · Zbl 0576.53036
[30] Lira, J. H.; Tojeiro, R.; Vitório, F., A Bonnet theorem for isometric immersions into products of space forms, Arch. Math., 95, 469-479 (2010) · Zbl 1208.53061
[31] Mafio, F.; Vitõr io, F., Minimal immersions of Riemannian manifolds in products of space forms, J. Math. Anal. Appl., 424, 260-268 (2015) · Zbl 1305.53064
[32] Mustafa, A.; Uddin, S.; Al-Solamy, F. R., Chen-Ricci inequality for warped products in Kenmotsu space forms and its applications, RACSAM, 113, 3585 (2019) · Zbl 1427.53075
[33] Myers, S. B., Riemannian manifolds with positive mean curvature, Duke Math. J., 8, 2, 401-404 (1941) · JFM 67.0673.01
[34] Nash, J., The imbedding problem for Riemannian manifolds, Ann. of Math., 63, 20-63 (1956) · Zbl 0070.38603
[35] Nolker, S., Isometric immersions of warped products, Differential Geom. Appl., 6, 1-30 (1996) · Zbl 0881.53052
[36] Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, 14, 33-340 (1962) · Zbl 0115.39302
[37] Palmer, B., The Gauss map of a spacelike constant mean curvature hypersurface of Minkowski space, Comment. Math. Helv., 65, 52-57 (1990) · Zbl 0702.53008
[38] Simons, J., Minimal varieties in Riemannian manifolds, Ann. Math., 88, 62-105 (1968) · Zbl 0181.49702
[39] Solomon, B., Harmonic maps to spheres, J. Differential Geom., 21, 151-162 (1985) · Zbl 0589.58035
[40] Takahashi, T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18, 380-385 (1966) · Zbl 0145.18601
[41] Yano, K.; Kon, M., Structures on Manifolds (1984), World Scientic · Zbl 0557.53001
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.