Pointwise semi-slant Riemannian maps into almost Hermitian manifolds and Casorati inequalities
Abstract
UDC 514
As a natural generalization of slant submanifolds [B.-Y. Chen, Bull. Austral. Math. Soc., 41, No. 1, 135 (1990)], slant submersions [B. Şahin, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 54, No. 102, 93 (2011)], slant Riemannian maps [B. Şahin, Quaestion. Math., 36, No. 3, 449 (2013) and Int. J. Geom. Methods Mod. Phys., 10, Article 1250080 (2013)], pointwise slant submanifolds [B.-Y. Chen, O. J. Garay, Turk. J. Math., 36, 630 (2012)], pointwise slant submersions [J. W. Lee, B. Şahin, Bull. Korean Math. Soc., 51, No. 4, 1115 (2014)], pointwise slant Riemannian maps [Y. Gündüzalp, M. A. Akyol, J. Geom. and Phys., 179, Article 104589 (2022)], semi-slant submanifolds [N. Papaghiuc, Ann. Ştiinƫ. Univ. Al. I. Cuza Iaṣi. Mat. (N.S.), 40, 55 (1994)], semi-slant submersions [K.-S. Park, R. Prasad, Bull. Korean Math. Soc., 50, No. 3, Article 951962 (2013)], and semi-slant Riemannian maps [K.-S. Park, B. Şahin, Czechoslovak Math. J., 64, No. 4, 1045 (2014)], we introduce a new class of Riemannian maps, which are called {\it pointwise semi-slant Riemannian maps,} from Riemannian manifolds to almost Hermitian manifolds. We first give some examples, present a characterization, and obtain the geometry of foliations in terms of the distributions involved in the definition of these maps. We also establish necessary and sufficient conditions for pointwise semi-slant Riemannian maps to be totally geodesic and harmonic, respectively. Finally, we determine the Casorati curvatures for pointwise semi-slant Riemannian maps in the complex space form.
References
R. Abraham, J. E. Marsden, T. Ratiu, Manifolds, tensor analysis and applications, Appl. Math. Sci., 75, Springer, New York (1988).
M. A. Akyol, R. Prasad, Semi-slant $ξ^⊥$, hemi-slant $ξ^⊥$-Riemannian submersions and quasi hemi-slant submanifolds, in: B. Y. Chen, M. H. Shahid, F. Al-Solamy (eds.), Contact Geometry of Slant Submanifolds, Springer, Singapore (2022), p. 301–332.
M. A. Akyol, Conformal semi-slant submersions, Int. J. Geom. Methods Mod. Phys., 14, № 7, Article 1750114 (2017).
M. A. Akyol, R. Sari, On semi-slant $ξ^⊥$-Riemannian submersions, Mediterr. J. Math., 14, Article 234 (2017); https://doi.org/10.1007/s00009-017-1035-2.
M. A. Akyol, B. Şahin, Conformal anti-invariant Riemannian maps to Kähler manifolds, U.P.B. Sci. Bull. Ser. A, 80, Issue 4 (2018).
M. A. Akyol, B. Şahin, Conformal semi-invariant Riemannian maps to Kähler manifolds, Rev. Union Mat. Argentina, 60, № 2, 459–468 (2019).
M. A. Akyol, B. Şahin, Conformal slant Riemannian maps to Kähler manifolds, Tokyo J. Math., 42, № 1, 225–237 (2019).
M. Aquib, J. W. Lee, G. E. Vǐlcu, D. W. Yoon, Classification of Casorati ideal Lagrangian submanifolds in complex space forms, Different. Geom. and Appl., 63, 30–49 (2019).
M. Aquib, M. H. Shahid, Generalized normalized $delta$-Casorati curvature for statistical submanifolds in quaternion Kähler-like statistical space forms, J. Geom., 109, № 1, Article 13 (2018).
P. Baird, J. C. Wood, Harmonic morphisms between Riemannian manifolds, Clarendon Press, Oxford (2003).
J. P. Bourguignon, H. B. Lawson, Stability and isolation phenomena for Yangmills fields, Commun. Math. Phys., 79, 189–230 (1981).
J. P. Bourguignon, H. B. Lawson, A mathematician's visit to Kaluza–Klein theory, Rend. Semin. Mat. Univ. Politec. Torino, special issue, 143–163 (1989).
P. Candelas, G. T. Horowitz, A. Strominger, E. Witten, Vacuum configurations for superstrings, Nuclear Phys. B, 258, 46–74 (1985).
J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, M. Fernandez, Slant submanifolds in Sasakian manifolds, Glasgow Math. J., 42, № 1, 125–138 (2000).
F. Casorati, Nuova defnizione della curvatura delle superfcie e suo confronto con quella di Gauss (New definition of the curvature of the surface and its comparison with that of Gauss), Rend. Inst. Mat. Accad. Lomb. Ser. II, 22, № 8, 335–346 (1889).
B.-Y. Chen, Slant immersions, Bull. Austral. Math. Soc., 41, № 1, 135–147 (1990).
B.-Y. Chen, O. J. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turk. J. Math., 36, 630–640 (2012).
F. Etayo, On quasi-slant submanifolds of an almost Hermitian manifold, Publ. Math. Debrecen, 53, 217–223 (1998).
E. Garcia-Rio, D. N. Kupeli, Semi-Riemannian maps and their applications, Kluwer Academic, Dordrecht (1999).
A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. and Mech., 16, 715–737 (1967).
Y. Gündüzalp, M. A. Akyol, Pointwise slant Riemannian maps from Kähler manifolds, J. Geom. and Phys., 179, Article 104589 (2022).
Y. Gündüzalp, M. A. Akyol, Remarks on conformal anti-invariant Riemannian maps to cosymplectic manifolds, Hacet. J. Math. and Stat., 50, № 4, 1131–1139 (2021).
M. Falcitelli, S. Ianus, A. M. Pastore, Riemannian submersions and related topics, World Sci. (2004).
A. E. Fischer, Riemannian maps between Riemannian manifolds, Contemp. Math., 132, 331–366 (1992).
C. W. Lee, J. W. Lee, G. E. Vǐlcu, Optimal inequalities for the normalized $delta$-Casorati curvatures of submanifolds in Kenmotsu space forms, Adv. Geom., 17, № 3, 355–362 (2017).
C. W. Lee, J. W. Lee, B. Şahin, G. E. Vilcu, Optimal inequalities for Riemannian maps and Riemannian submersions involving Casorati curvatures, Ann. Mat. Pura ed Appl., 200, 1277–1295 (2021).
J. W. Lee, B. Şahin, Pointwise slant submersions, Bull. Korean Math. Soc., 51, № 4, 1115–1126 (2014).
S. Ianus, M. Visinescu, Kaluza–Klein theory with scalar fields and generalized Hopf manifolds, Class. Quantum Gravity, 4, 1317–1325 (1987).
S. Ianus, M. Visinescu, Space-time compactication and Riemannian submersions, in: G. Rassias (ed.), The Mathematical Heritage of C. F. Gauss, World Sci., River Edge (1991), p. 358–371.
M. T. Mustafa, Applications of harmonic morphisms to gravity, J. Math. Phys., 41, 6918–6929 (2000).
T. Nore, Second fundamental form of a map, Ann. Mat. Pure Appl., 146, 281–310 (1987).
B. O'Neill, The fundamental equations of a submersion, Michigan Math. J., 13, 458–469 (1966).
N. Papaghiuc, Semi-slant submanifolds of a Kählerian manifold, Ann. Ştiinƫ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 40, 55–61 (1994).
K.-S. Park, R. Prasad, Semi-slant submersions, Bull. Korean Math. Soc., 50, № 3, 951–962 (2013).
K.-S. Park, B. Şahin, Semi-slant Riemannian maps into almost Hermitian manifolds, Czechoslovak Math. J., 64, № 4, 1045–1061 (2014).
K.-S. Park, Almost $h$-semi-slant Riemannian maps, Taiwanese J. Math., 17, № 3, 937–956 (2013).
K.-S. Park, Pointwise slant and pointwise semi-slant submanifolds in almost contact metric manifolds, Mathematics, 8, Article 985 (2020).
K.-S. Park, Pointwise almost $h$-semi-slant submanifolds, Int. J. Math., 26, № 11, Article 1550099 (2015).
K.-S. Park, On the pointwise slant submanifolds, in: Y. Suh, Y. Ohnita, J. Zhou, B. Kim, H. Lee (eds.), Hermitian–Grassmannian Submanifolds, Springer Proc. Math. and Stat., 203, Springer, Singapore (2017).
R. Prasad, S. Pandey, Slant Riemannian maps from an almost contact manifold, Filomat, 31, № 13, 3999–4007 (2017).
C. Sayar, F. Özdemir, H. M. Taştan, Pointwise semi-slant submersions whose total manifolds are locally product Riemannian manifolds, Int. J. Maps in Math., 1, Issue 1, 91–115 (2018).
S. A. Sepet, M. Ergüt, Pointwise slant submersions from cosymplectic manifolds, Turk. J. Math., 40, 582–593 (2016).
B. Şahin, Conformal Riemannian maps between Riemannian manifolds, their harmonicity and decomposition theorems, Acta Appl. Math., 109, № 3, 829–847 (2010).
B. Şahin, Invariant and anti-invariant Riemannian maps to Kähler manifolds, Int. J. Geom. Methods Mod. Phys., 7, № 3, 1–19 (2010).
B. Şahin, Slant Riemannian maps from almost Hermitian manifolds, Quaestiones Math., 36, № 3, 449–461 (2013).
B. Şahin, Slant Riemannian maps to Kähler manifolds, Int. J. Geom. Methods Mod. Phys., 10, Article 1250080 (2013).
B. Şahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 54, № 102, 93–105 (2011).
B. Şahin, Riemannian submersions, Riemannian maps in Hermitian geometry, and their applications, Elsevier, Academic Press (2017).
B. Şahin, Warped product pointwise semi-slant submanifolds of Kähler manifolds, Port. Math., 70, 252–268 (2013).
A. J. Tromba, Teichmüller theory in Riemannian geometry: based on lecture notes by Jochen Denzler, Lect. in Math. ETH Zürich, Birkhäuser, Basel (1992).
M. M. Tripathi, Inequalities for algebraic Casorati curvatures and their applications, Note Mat., 37, suppl. 1, 161–186 (2017).
G. E. Vilcu, An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvatures, J. Math. Anal. and Appl., 465, № 2, 1209–1222 (2018).
B. B. Watson, $G,$ $G'$-Riemannian submersions and nonlinear gauge field equations of general relativity, in: T. Rassias (ed.), Global Analysis – Analysis on Manifolds, dedicated M. Morse, Teubner-Texte Math., 57, Teubner, Leipzig (1983), p. 324–349.
K. Yano, M. Kon, Structures on manifolds, World Sci. (1985).
L. Zhang, X. Pan, P. Zhang, Inequalities for Casorati curvature of Lagrangian submanifolds in complex space forms, Adv. Math. (China), 45, № 5, 767–777 (2016).
P. Zhang, L. Zhang, Inequalities for Casorati curvatures of submanifolds in real space forms, Adv. Geom., 16, № 3, 329–335 (2016).
Copyright (c) 2024 Mehmet Akyol
This work is licensed under a Creative Commons Attribution 4.0 International License.