×

Chen’s first inequality for Riemannian maps. (English) Zbl 1357.53024

Summary: We obtain a basic Chen inequality for Riemannian maps between Riemannian manifolds.

MSC:

53B20 Local Riemannian geometry
53B21 Methods of local Riemannian geometry
Full Text: DOI

References:

[1] [AMR]R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer, New York, 1988. · Zbl 0875.58002
[2] [ACM]P. Alegre, B. Y. Chen and M. I. Munteanu, Riemannian submersions, {\(\delta\)}invariants, and optimal inequality, Ann. Global Anal. Geom. 42 (2012), 317– 331. · Zbl 1253.53057
[3] [BW]P. Baird and J. C. Wood, Harmonic Morphisms Between Riemannian Manifolds, Clarendon Press, Oxford, 2003. · Zbl 1055.53049
[4] [C1]B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel) 60 (1993), 568–578. · Zbl 0811.53060
[5] [C2]B. Y. Chen, A general inequality for submanifolds in complex space forms and its applications, Arch. Math. (Basel) 67 (1996), 519–528. · Zbl 0871.53043
[6] [C3]B. Y. Chen, Riemannian submanifolds, in: Handbook of Differential Geometry, Vol. I, North-Holland, Amsterdam, 2000, 187–418.
[7] [C4]B. Y. Chen, Riemannian submersions, minimal immersions and cohomology class, Proc. Japan Acad. Ser A 81 (2005), 162–167. · Zbl 1147.53312
[8] [C5]B. Y. Chen, Examples and classification of Riemannian submersions satisfying a basic equation, Bull. Austral. Math. Soc. 72 (2005), 391–402. · Zbl 1093.53064
[9] [C6]B. Y. Chen, Pseudo-Riemannian Geometry, {\(\delta\)}-invariants and Applications, World Sci., 2011.
[10] [FG]M. Faghfouri and N. Ghaffarzadeh, Chen’s inequality for invariant submanifolds in generalized ({\(\kappa\)}, {\(\mu\)})-space forms, Global J. Adv. Res. Class. Modern Geom. 4 (2015), 86–101.
[11] [FIP]M. Falcitelli, S. Ianus and A. M. Pastore, Riemannian Submersions and Related Topics, World Sci., 2004. · Zbl 1067.53016
[12] [F]A. E. Fischer, Riemannian maps between Riemannian manifolds, in: Contemp. Math. 132, Amer. Math. Soc., 1992, 331–366. · Zbl 0780.53033
[13] [GRK]E. Garc’ıa-R’ıo and D. N. Kupeli, Semi-Riemannian Maps and Their Applications, Kluwer, 1999. · Zbl 0924.53003
[14] [GKKT]M. G”ulbahar, E. Kılı\c{}c, S. Kele\c{}s and M. M. Tripathi, Some basic inequalities for submanifolds of nearly quasi-constant curvature manifolds, Differential Geom. Dynam. Systems 16 (2014), 156–167.
[15] [G]R. S. Gupta, B. Y. Chen’s inequalities for bi-slant submanifolds in cosymplectic space forms, Sarajevo J. Math. 9 (21) (2013), 117–128. · Zbl 1293.53066
[16] [KTG]E. Kılı\c{}c, M. M. Tripathi and M. G”ulbahar, Chen–Ricci inequalities for submanifolds of Riemannian and Kaehlerian product manifolds, Ann. Polon. Math. 116 (2016), 37–56. 258B. S.ahin · Zbl 1337.53069
[17] [LLY]C. W. Lee, J. W. Lee and D. W. Yoon, Improved Chen inequality of Sasakian space forms with the Tanaka–Webster connection, Filomat 29 (2015), 1525– 1533. · Zbl 1474.53231
[18] [MR]A. Mihai and I. N. Radulescu, Scalar and Ricci curvatures of special contact slant submanifolds in Sasakian space forms, Adv. Geom. 14 (2014), 147–159. · Zbl 1334.53052
[19] [N]T. Nore, Second fundamental form of a map, Ann. Mat. Pura Appl. 146 (1987), 281–310. · Zbl 0631.53015
[20] [O]B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469.
[21] [OD]C. ”Ozg”ur and A. De, Chen inequalities for submanifolds of a Riemannian manifold of nearly quasi-constant curvature, Publ. Math. Debrecen 82 (2013), 439– 450. · Zbl 1299.53123
[22] [OM]C. ”Ozg”ur and A. Mihai, Chen inequalities for submanifolds of real space forms with a semi-symmetric non-metric connection, Canad. Math. Bull. 55 (2012), 611–622.
[23] [S1]B. S.ahin, Invariant and anti-invariant Riemannian maps to K”ahler manifolds, Int. J. Geom. Methods Modern Phys. 7 (2010), 337–355. · Zbl 1193.53148
[24] [S2]B. S.ahin, Semi-invariant Riemannian maps to K”ahler manifolds, Int. J. Geom. Methods Modern Phys. 7 (2011), 1439–1454. · Zbl 1242.53039
[25] [S3]B. S.ahin, A survey on differential geometry of Riemannian maps between Riemannian manifolds, Ann. Alexandru Ioan Cuza Univ. Math., DOI: 10.1515/aicu2015-0001.
[26] [S]P. Sol’orzano, Group norms and their degeneration in the study of parallelism, PhD thesis, Stony Brook Univ., 2011.
[27] [V]G. E. Vılcu, On Chen invariants and inequalities in quaternionic geometry, J. Inequal. Appl. 2013, 2013:66, 14 pp. · Zbl 1282.53043
[28] [ZZ]L. Zhang and P. Zhang, Notes on Chen’s inequalities for submanifolds of real space forms with a semi-symmetric non-metric connection, J. East China Norm. Univ. Natur. Sci. Ed. 2015, no. 1, 6–15.
[29] [Z]P. Zhang, Remarks on Chen’s inequalities for submanifolds of a Riemannian manifold of nearly quasi-constant curvature, Vietnam J. Math. 43 (2015), 557– 569. · Zbl 1337.53075
[30] [ZZS]P. Zhang, L. Zhang and W. Song, Chen’s inequalities for submanifolds of a Riemannian manifold of quasi-constant curvature with a semi-symmetric metric connection, Taiwanese J. Math. 18 (2014), 1841–1862. Bayram S.ahin Department of Mathematics Ege University 35100, Izmir, Turkey E-mail: bayram.sahin@ymail.com 1 Introduction2 Preliminaries3 Chen inequality for Riemannian mapsReferences · Zbl 1357.53063
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.