Chen-Type Inequality for Generic Submanifolds of Quaternionic Space Form and Its Application

Year 2024, , 90 - 98, 30.09.2024

Amine Yılmaz

https://doi.org/10.53570/jnt.1527579

Abstract

In 1993, the theory of Chen invariants started when Chen wrote basic inequalities for submanifolds in space forms. This inequality is called Chen’s first inequality. Afterward, many geometers studied many papers dealing with this new inequality. The present paper aims to establish a Chen inequality for quaternionic generic submanifolds in a quaternionic space form and obtain this inequality for real hypersurfaces.

Keywords

Chen-type inequality, generic submanifold, quaternionic space form, real hypersurfaces

References

• A. Bejancu, Geometry of CR-submanifolds, Kluwer, Dortrecht, 1986.
• B. Y. Chen, On CR-submanifolds of Kaehler manifold I, Journal of Differential Geometry 16 (2) (1981) 305-322.
• S. Dragomir, M. H. Shahid, F. Al-Solamy, S. M. Hasan, Geometry of Cauchy-Riemann submanifolds, Springer, Singapore, 2016.
• B. Y. Chen, Cohomology of CR-submanifolds, Annales de la Faculté des Sciences de Toulouse Mathématiques 3 (2) (1981) 167-172.
• B. Şahin, F. Şahin, Homology of contact CR-warped product submanifolds of an odd-dimensional unit sphere, Bulletin of the Korean Mathematical Society 52 (1) (2015) 215-222.
• F. Şahin, On the topology of CR-warped product submanifolds, International Journal of Geometric Methods in Modern Physics 15 (02) (2018) 1850032 7 pages.
• F. Şahin, Homology of submanifolds of six dimensional sphere, Journal of Geometry and Physics 145 (2019) 103471 6 pages.
• M. Barros, F. Urbano, Quaternion CR-submanifolds of quaternion manifolds, Kodai Mathematical Journal 4 (3) (1981) 399-417.
• E. Abedi, G. Haghighatdoost, M. Ilmakchi, Z. Nazari, Contact 3-structure QR-warped product submanifold in Sasakian space form, Turkish Journal of Mathematics 37 (2) (2013) 340-347.
• A. Bejancu, H. R. Farran, On totally umbilical QR-submanifolds of quaternion Kaehlerian manifolds, Bulletin of the Australian Mathematical Society 62 (1) (2000) 95-103.
• R. Güneş, B. Şahin, S. Keleş, QR-submanifolds and almost contact 3-Structure, Turkish Journal of Mathematics 24 (3) (2000) 239-250.
• H. S. Kim, J. S. Pak, Certain QR-submanifolds of maximal QR-dimension in a quaternionic space form, Balkan Journal of Geometry and its Applications 18 (1) (2013) 31-46.
• G. Macsim, A. Mihai, A $\delta$-invariant for QR-submanifolds in quaternion space forms, International Electronic Journal of Geometry 11 (2) (2018) 8-17.
• G. Macsim, A. Mihai, I. Mihai, $\delta(2,2)$-invariant for Lagrangian submanifolds in quaternionic space forms, Mathematics 8 (4) (2020) 480.
• Z. Nazari, E. Abedi, Ricci solitons on QR-hypersurfaces of a quaternionic space form $\mathbb{Q}^n$, Acta Universitatis Sapientiae, Mathematica 8 (2) (2016) 271-281.
• B. Şahin, R. Güneş, On some properties of distributions in QR-submanifolds, Hacettepe Journal of Mathematics and Statistics 28 (1999) 15-23.
• B. Şahin, On QR-submanifolds of quaternionic space form}, Turkish Journal of Mathematics 25 (3) (2001) 413-425.
• B. Şahin, R. Güneş, QR-submanifolds of a locally conformal quaternion Kaehler manifold, Publicationes Mathematicae Debrecen 63 (2003) 157-174.
• B. Y. Chen, Differential geometry of real submanifolds in a Kaehler manifold, Monatshefte für Mathematik 91 (1981) 257-274.
• Y. W. Choe, U. H. Ki, R. Takagi, Compact minimal generic submanifolds with parallel normal section in a complex projective space, Osaka Journal of Mathematics 37 (2) (2000) 489-499.
• U. C. De, A. K. Sengupta, C. Calin, Generic submanifolds of quasi-Sasakian manifolds, Demonstratio Mathematica 37 (2) (2004) 429-437.
• U. C. De, A. K. Sengupta, Generic submanifolds of a Lorentzian para-Sasakian manifold, Soochow Journal of Mathematics 27 (1) (2001) 29-36.
• M. Kon, On minimal generic submanifolds immersed in $S^{2m+1}$, Colloquium Mathematicum 90 (2) (2001) 299-304.
• G. E. Vilcu, On generic submanifolds of manifolds endowed with metric mixed 3-structures, Communications in Contemporary Mathematics 18 (6) (2016) 1550081 21 pages.
• A. Yılmaz, D. Poyraz, B. Şahin, Generic QR-submanifolds of quaternion Kaehler manifolds, Acta Mathematica Universitatis Comenianae 90 (3) (2021) 341-351.
• A. Yılmaz, Generic submanifolds of quaternion Kaehler manifolds and Wintgen type inequality, Filomat 38 (12) (2024) 4167-4185.
• F. Al-Solamy, B. Y. Chen, S. Deshmukh, Two optimal inequalities for anti-holomorphic submanifolds and their applications, Taiwanese Journal of Mathematics 18 (1) (2014) 199-217.
• M. D. Siddiqi, A. N. Siddiqui, O. Bahadir, Generalized Wintgen inequalities for submanifolds of trans-Sasakian space form, Matimyás Matematika 44 (1) (2021) 1-14.
• A. N. Siddiqui, Y. J. Suh, O. Bahadır, Extremities for statistical submanifolds in Kenmotsu statistical manifolds, Filomat 35 (2) (2021) 591-603.
• A. Bejancu, QR-submanifolds of quaternion Kaehlerian manifolds, Chinese Journal of Mathematics 14 (2) (1986) 81-94.
• B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Archiv der Mathematik 60 (1993) 568-578.
• B. Y. Chen, An optimal inequality for $CR$-warped products in complex space forms involving $CR$ $\delta$-invariant, International Journal of Mathematics 23 (3) (2012) 17 pages.