Abstract
The aim of the present paper is to define and study semi-slant \(\xi ^\perp \)-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds as a generalization of anti-invariant \(\xi ^\perp \)-Riemannian submersions, semi-invariant \(\xi ^\perp \)-Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base manifold to be a locally product manifold and proving new conditions to be totally umbilical and totally geodesicness, respectively. Moreover, some examples of such submersions are mentioned.
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References
Akyol, M.A.: Conformal semi-slant submersions. Int. J. Geom. Methods Mod. Phys. 14(7), 1750114 (2017)
Akyol, M.A.: Conformal anti-invariant submersions from cosymplectic manifolds. Hacettepe J. Math. Stat. 46(2), 177–192 (2017)
Akyol, M.A., Gündüzalp, Y.: Hemi-slant submersions from almost product Riemannian manifolds. Gulf J. Math. 4(3), 15–27 (2016)
Akyol, M.A., Sarı, R., Aksoy, E.: Semi-invariant \(\xi ^\perp -\)Riemannian submersions from almost contact metric manifolds. Int. J. Geom. Methods Mod. Phys. 14(5), 1750074 (2017)
Ali, S., Fatima, T.: Anti-invariant Riemannian submersions from nearly Kaehler manifolds. Filomat 27(7), 1219–1235 (2013)
Baird, P., Wood, J.C.: Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, vol. 29. Oxford University Press, The Clarendon Press, Oxford (2003)
Blair, D.E.: Contact Manifold in Riemannain Geometry. Lecture Notes in Math, vol. 509. Springer, Berlin (1976)
Bourguignon, J.P., Lawson, H.B.: Stability and isolation phenomena for Yang-mills fields. Commun. Math. Phys. 79, 189–230 (1981)
Bourguignon, J.P., Lawson, H.B.: A mathematician’s Visit to Kaluza-Klein Theory, pp. 143–163. Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1989)
Cabrerizo, J.L., Carriazo, A., Fernandez, L.M., Fernandez, M.: Semi-slant submanifolds of a Sasakian manifold. Geometriae Dedicata 78(2), 183–199 (1999)
Chinea, D.: Almost contact metric submersions. Rend. Circ. Mat. Palermo 34(1), 89–104 (1985)
Erken, I.K., Murathan, C.: Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian submersions. Filomat 29(7), 1429–1444 (2015)
Erken, I.K., Murathan, C.: Slant Riemannian submersions from Sasakian manifolds. Arap J. Math. Sci. 22(2), 250–264 (2016)
Erken, I.K., Murathan, C.: On slant Riemannian submersions for cosymplectic manifolds. Bull. Korean Math. Soc. 51(6), 1749–1771 (2014)
Falcitelli, M., Ianus, S., Pastore, A.M.: Riemannian Submersions and Related Topics. World Scientific, River Edge (2004)
Gray, A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16, 715–737 (1967)
Gündüzalp, Y.: Anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds. J. Funct. Spaces Appl. 2013, Article ID 720623 (2013)
Gündüzalp, Y.: Slant submersions from almost product Riemannian manifolds. Turk. J. Math. 37, 863–873 (2013)
Gündüzalp, Y.: Semi-slant submersions from almost product Riemannian manifolds. Demonstratio Mathematica 49(3), 345–356 (2016)
Lee, J.W.: Anti-invariant \(\xi ^{\perp }-\)Riemannian submersions from almost contact manifolds. Hacettepe J. Math. Stat. 42(3), 231–241 (2013)
Lee, J.W., Şahin, B.: Pointwise slant submersions. Bull. Korean Math. Soc. 51(4), 1115–1126 (2014)
Ianus, S., Visinescu, M.: Kaluza-Klein theory with scalar fields and generalized Hopf manifolds. Class. Quantum Gravity 4, 1317–1325 (1987)
Ianus, S., Visinescu, M.: Space-time compactication and Riemannian submersions. In: Rassias, G. (ed.) The Mathematical Heritage of C. F. Gauss, pp. 358–371. World Scientic, River Edge (1991)
Ianus, S., Mazzocco, R., Vilcu, G.E.: Riemannian submersions from quaternionic manifolds. Acta Appl. Math. 104(1), 83–89 (2008)
Mustafa, M.T.: Applications of harmonic morphisms to gravity. J. Math. Phys. 41, 6918–6929 (2000)
O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 458–469 (1966)
Park, K.S.: h-semi-invariant submersions. Taiwan. J. Math. 16(5), 1865–1878 (2012)
Park, K.S.: H-V-semi-slant submersions from almost quaternionic Hermitian manifolds. Bull. Korean Math. Soc. 53(2), 441–460 (2016)
Park, K.S., Prasad, R.: Semi-slant submersions. Bull. Korean Math. Soc. 50(3), 951–962 (2013)
Park, K.S.: h-slant submersions. Bull. Korean Math. Soc. 49(2), 329–338 (2012)
Sasaki, S., Hatakeyama, Y.: On differentiable manifolds with contact metric structure. J. Math. Soc. Japan 14, 249–271 (1961)
Sepet, S.A., Ergut, M.: Pointwise slant submersions from cosymplectic manifolds. Turk. J. Math. 40, 582–593 (2016)
Ponge, R.: H. Reckziegel., Twisted products in pseudo-Riemannian geometry. Geom. Dedicata. 48(1), 15–25 (1993)
Şahin, B.: Anti-invariant Riemannian submersions from almost Hermitian manifolds. Central Eur. J. Math. (3), 437–447 (2010)
Şahin, B.: Semi-invariant Riemannian submersions from almost Hermitian manifolds. Can. Math. Bull. 56, 173–183 (2011)
Şahin, B.: Slant submersions from almost Hermitian manifolds. Bull. Math. Soc. Sci. Math. Roumanie. 1, 93–105 (2011)
Şahin, B.: Riemannian submersions from almost Hermitian manifolds. Taiwan. J. Math. 17(2), 629–659 (2013)
Şahin, B.: Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications. Elsevier, Academic, Amsterdam (2017)
Tastan, H.M., Şahin, B., Yanan, Ş.: Hemi-slant submersions. Mediterr. J. Math. 13(4), 2171–2184 (2016)
Vilcu, G.E.: Mixed paraquaternionic 3-submersions. Indag. Math. (N.S.) 24(2), 474–488 (2013)
Vilcu, A.D., Vilcu, G.E.: Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions. Entropy 17(9), 6213–6228 (2015)
Watson, B.: Almost Hermitian submersions. J. Differ. Geom. 11(1), 147–165 (1976)
Taştan, H.M.: Lagrangian submersions. Hacettepe J. Math. Stat. 43(6), 993–1000 (2014)
Taştan, H.M.: Lagrangian submersions from normal almost contact manifolds. Filomat 31(12), 3885–3895 (2017)
Watson, B.: G, G’-Riemannian submersions and nonlinear gauge field equations of general relativity. In: Rassias, T. (ed.) Global Analysis—Analysis on manifolds, dedicated M. Morse. Teubner-Texte Math., vol. 57, pp. 324–349. Teubner, Leipzig (1983)
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Akyol, M.A., Sarı, R. On Semi-Slant \({\varvec{\xi ^\perp }}\)-Riemannian Submersions. Mediterr. J. Math. 14, 234 (2017). https://doi.org/10.1007/s00009-017-1035-2
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DOI: https://doi.org/10.1007/s00009-017-1035-2
Keywords
- Riemannian submersion
- Sasakian manifold
- anti-invariant \(\xi ^\perp \)-Riemannian submersion
- semi-invariant \(\xi ^\perp \)-Riemannian submersion
- slant Riemannian submersion