Pointwise hemi-slant Riemannian maps ($\mathcal{PHSRM}$) from almost Hermitian manifolds

Year 2024, Volume: 53 Issue: 5, 1218 - 1237, 15.10.2024

Mehmet Akif Akyol , Yılmaz Gündüzalp

https://doi.org/10.15672/hujms.1219010

Abstract

In 2022, the notion of pointwise slant Riemannian maps were introduced by Y. Gündüzalp and M. A. Akyol in [J. Geom. Phys. {179}, 104589, 2022] as a natural generalization of slant Riemannian maps, slant Riemannian submersions, slant submanifolds. As a generalization of pointwise slant Riemannian maps and many subclasses notions, we introduce pointwise hemi-slant Riemannian maps (briefly, $\mathcal{PHSRM}$) from almost Hermitian manifolds to Riemannian manifolds, giving a figure which shows the subclasses of the map and a non-trivial (proper) example and investigate some properties of the map, we deal with their properties: the J-pluriharmonicity, the J-invariant, and the totally geodesicness of the map. Finally, we study some curvature relations in complex space form, involving Chen inequalities and Casorati curvatures for $\mathcal{PHSRM}$, respectively.

Keywords

Riemannian map, Hermitian manifold, slant Riemannian map, hemi-slant submersion, hemi-slant Riemannian map, pointwise hemi-slant Riemannian map

Supporting Institution

TUBİTAK

Project Number

121F277

Thanks

This paper is supported by 1001-Scientific and Technological Research Projects Funding Program of The Scientific and Technological Research Council of Turkey (TUBITAK) with project number 121F277.

References

• [1] R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications, Applied Mathematical Sciences, Vol. 75, Springer, New York, 1988.
• [2] M.A. Akyol, On Pointwise Riemannian Maps in Complex Geometry, International Symposium on Differential Geometry and Its Applications, Maulana Azad National Urdu University, Gachibowli, Hyderabad - 500032, Telangana, India, 2022.
• [3] M.A. Akyol and Y. Gündüzalp, Pointwise slant Riemannian maps (PSRM) to almost Hermitian manifolds, Mediterr. J. Math. 20, 116, 2023.
• [4] M.A. Akyol and B. Şahin, Conformal anti-invariant Riemannian maps to Kaehler manifolds, U.P.B. Sci. Bull., Series A 80 (4), 2018.
• [5] M.A. Akyol and B. Şahin, Conformal semi-invariant Riemannian maps to Kaehler manifolds, Rev. Un. Mat. Argentina 60 (2), 459–468, 2019.
• [6] M.A. Akyol and B. Şahin, Conformal slant Riemannian maps to Kaehler manifolds, Tokyo J. Math. 42 (1), 225-237, 2019.
• [7] M. Aquib, J.W. Lee, G. E. Vilcu and D. W. Yoon, Classifcation of Casorati ideal Lagrangian submanifolds in complex space forms, Differ. Geom. Appl. 63, 30–49, 2019.
• [8] M. Aquib and M. H. Shahid, Generalized normalized δ-Casorati curvature for statistical submanifolds in quaternion Kaehler-like statistical space forms, J. Geom. 109 (1), Art. 13, 2018.
• [9] M.E. Aydın, A. Mihai and I. Mihai, Some Inequalities on submanifolds in statistical manifolds of constant curvature, Filomat 29 (3), 465–477, 2015.
• [10] P. Baird and J.C. Wood, Harmonic Morphisms Between Riemannian Manifolds, Clarendon Press, Oxford, 2003.
• [11] J.P. Bourguignon and H.B. Lawson, Stability and isolation phenomena for Yangmills fields, Commun. Math. Phys. 79, 189–230, 1981.
• [12] J.P. Bourguignon and H.B. Lawson, A mathematicians Visit to Kaluza-Klein Theory, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue, 143–163, 1989.
• [13] J.L. Cabrerizo, A. Carriazo, L.M. Fernandez and M. Fernandez, Slant submanifolds in Sasakian manifolds, Glasgow Math. J. 42 (1), 125–138, 2000.
• [14] F. Casorati, Nuova defnizione della curvatura delle superfcie e suo confronto con quella di Gauss. (New defnition of the curvature of the surface and its comparison with that of Gauss). Rend. Inst. Matem. Accad. Lomb. Ser. II 22 (8), 335–346, 1889.
• [15] B.-Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Leuven, 1990.
• [16] B.-Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel) 60, 568–578, 1993.
• [17] B.-Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasg. Math. J. 41 (1), 33–41, 1999.
• [18] B. Y. Chen and O. J. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turk. J. Math. 36, 630–640, 2012.
• [19] F. Etayo, On quasi-slant submanifolds of an almost Hermitian manifold, Publ. Math. Debrecen 53, 217–223, 1998.
• [20] E. Garcia-Rio and D. N. Küpeli, Semi-Riemannian Maps and Their Applications, Kluwer Academic, Dordrecht, 1999.
• [21] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16, 715–737, 1967.
• [22] M. Gülbahar, Ş.E. Meriç and E. Kılıç, Sharp inequalities involving the Ricci curvature for Riemannian submersions, Kragujevac J. Math. 41 (2), 279–293, 2017.
• [23] Y. Gündüzalp and M.A. Akyol, Remarks on conformal anti-invariant Riemannian maps to cosymplectic manifolds, Hacet. J. Math. Stat. 50 (4), 1131–1139, 2021.
• [24] Y. Gündüzalp and M.A. Akyol, Pointwise slant Riemannian maps from Kaehler manifolds, J. Geom. Phys. 179, 104589, 2022.
• [25] Y. Gündüzalp and M.A. Akyol, Pointwise semi-slant Riemannian (PSSR) maps from almost Hermitian manifolds, Filomat 37 (13), 4271–4286, 2023.
• [26] M. Falcitelli, S. Ianus and A. M. Pastore, Riemannian Submersions and Related Topics, World Scientific, 2004.
• [27] A.E. Fischer, Riemannian maps between Riemannian manifolds, Contemp. Math. 132, 331–366, 1992.
• [28] S. Ianus and M. Visinescu, Kaluza-Klein theory with scalar fields and generalized Hopf manifolds, Class. Quantum Grav. 4, 1317–1325, 1987.
• [29] S. Ianus and M. Visinescu, Space-time compactication and Riemannian submersions In: G. Rassias (ed.), The Mathematical Heritage of C. F. Gauss, 358–371. World Scientific, River Edge, 1991.
• [30] C.W. Lee, J.W. Lee, B. Şahin and G.E. Vilcu, Optimal inequalities for Riemannian maps and Riemannian submersions involving Casorati curvatures, Ann. Mat. Pura Appl. (1923 -) 200, 1277–1295, 2021.
• [31] C.W. Lee, J.W. Lee and G.E. Vilcu, Optimal inequalities for the normalized δ- Casorati curvatures of submanifolds in Kenmotsu space forms, Adv. Geom. 17 (3), 355–362, 2017.
• [32] J. Lee, J.H. Park, B. Şahin and D.Y. Song, Einstein conditions for the base of antiinvariant Riemannian submersions and Clairaut submersions, Taiwan. J. Math. 19 (4), 1145–1160, 2015.
• [33] J.W. Lee and B.S. ahin, Pointwise slant submersions, Bull. Korean Math. Soc. 51 (4), 1115–1126, 2014.
• [34] A. Mihai and I. Mihai, Curvature invariants for statistical submanifolds of Hessian manifolds of constant Hessian curvature, Mathematics 6, 44, 2018.
• [35] A. Mihai and C. Özgür, Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection, Taiwanese J. Math. 14 4, 1465–1477, 2010.
• [36] M.T. Mustafa, Applications of harmonic morphisms to gravity, J. Math. Phys. 41, 6918–6929, 2000.
• [37] B. O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13, 458–469, 1966.
• [38] K.S. Park, Almost h-semi-slant Riemannian maps, Taiwanese J. Math. 17 (3), 937– 956, 2013.
• [39] K.S. Park and B. Şahin, Semi-slant Riemannian maps into almost Hermitian manifolds, Czechoslovak Math. J. 64 (4), 1045–1061, 2014.
• [40] R. Prasad and S. Pandey, Slant Riemannian maps from an almost contact manifold, Filomat 31 (13), 3999-4007, 2017.
• [41] S.A. Sepet and H.G. Bozok, Pointwise semi-slant submersion, Differ. Geom. Dyn. Syst. 22, 1–10, 2020.
• [42] S.A. Sepet and M. Ergüt, Pointwise slant submersions from cosymplectic manifolds, Turk. J. Math. 40 (3), 582–593, 2016.
• [43] S.A. Sepet and M. Ergüt, Pointwise bi-slant submersions from cosymplectic manifolds, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69 (2), 1310–1319, 2020.
• [44] S.A. Sepet and M. Ergüt, Pointwise slant submersions from almost product Riemannian manifolds, J. Interdiscip. Math. 23 (3), 639–655, 2020.
• [45] B. Şahin, Conformal Riemannian maps between Riemannian manifolds,their harmonicity and decomposition theorems, Acta Appl. Math. 109 (3), 829–847, 2010.
• [46] B. Şahin, Invariant and anti-invariant Riemannian maps to Kahler manifolds, Int. J. Geom. Meth. Mod. Phys. 7 (3), 1–19, 2010.
• [47] B. Şahin, Slant Riemannian maps from almost Hermitian manifolds, Quaest. Math. 36 (3), 449–461, 2013.
• [48] B. Şahin, Slant Riemannian maps to Kaehler manifolds, Int. J. Geom. Meth. Mod. Phys. 10, 1250080, 2013.
• [49] B. Şahin, Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications, Elsevier, Academic Press, 2017.
• [50] B. Şahin, Hemi-slant Riemannian maps, Mediterr. J. Math. 14, Art. No: 10, 2017.
• [51] B. Şahin, Chens first inequality for Riemannian maps, Ann. Polon. Math. 117 (3) 249–258, 2016.
• [52] B. Şahin and Ş. Yanan, Conformal Riemannian maps from almost Hermitian manifolds, Turk. J. Math. 42, 2436–2451, 2018.
• [53] H.M. Taştan, B. Şahin and Ş. Yanan, Hemi-Slant Submersions, Mediterr. J. Math. 13, 2171–2184, 2016.
• [54] M.M. Tripathi, Inequalities for algebraic Casorati curvatures and their applications, Note Mat. 37 (1), 161–186, 2017.
• [55] G.E. Vilcu, B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms, Turk. J. Math. 34, 115–128, 2010.
• [56] G.E. Vilcu, On Chen invariants and inequalities in quaternionic geometry, J. Inequal. Appl. 2013, Art. No: 66, 2013.
• [57] G.E. Vilcu, An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvatures, J. Math. Anal. Appl. 465 (2), 1209–1222, 2018.
• [58] B. Watson, G, G-Riemannian submersions and nonlinear gauge field equations of general relativity, in: T. Rassias (ed.) Global AnalysisAnalysis on manifolds, dedicated M. Morse. Teubner-Texte Math., 57, 324–349, Teubner, Leipzig, 1983.
• [59] K. Yano and M. Kon, Structures on manifolds, World Scientific, 1985.
• [60] L. Zhang, X. Pan and P. Zhang, Inequalities for Casorati curvature of Lagrangian submanifolds in complex space forms, Adv. Math. (China) 45 (5), 767–777, 2016.
• [61] P. Zhang and L. Zhang, Inequalities for Casorati curvatures of submanifolds in real space forms, Adv. Geom. 16 (3), 329–335, 2016.