Quasi bi-slant submersions in contact geometry

Rajendra Prasad; Mehmet Akif Akyol; Sushil Kumar; Punit Kumar Singh

doi:10.4067/S0719-06462022000100001

Rajendra Prasad Department of Mathematics and Astronomy, University of Lucknow, Lucknow, India.
Mehmet Akif Akyol Department of Mathematics, Faculty of Sciences and Arts, Bingöl University, 12000, Bingöl, Turkey.
Sushil Kumar Department of Mathematics, Shri Jai Narain Post Graduate College, Lucknow, India.
Punit Kumar Singh Department of Mathematics and Astronomy, University of Lucknow, Lucknow, India.

DOI: https://doi.org/10.4067/S0719-06462022000100001

Keywords: Riemannian submersion, semi-invariant submersion, bi-slant submersion, quasi bi-slant submersion, horizontal distribution

Abstract

The aim of the paper is to introduce the concept of quasi bi-slant submersions from almost contact metric manifolds onto Riemannian manifolds as a generalization of semi-slant and hemi-slant submersions. We mainly focus on quasi bi-slant submersions from cosymplectic manifolds. We give some non-trivial examples and study the geometry of leaves of distributions which are involved in the definition of the submersion. Moreover, we find some conditions for such submersions to be integrable and totally geodesic.

References

M. A. Akyol, “Conformal anti-invariant submersions from cosymplectic manifolds”, Hacet. J. Math. Stat., vol. 46, no. 2, pp. 177–192, 2017.

M. A. Akyol and B. Şahin, “Conformal slant submersions”, Hacet. J. Math. Stat., vol. 48, no.1, pp. 28–44, 2019.

M. A. Akyol, “Conformal semi-slant submersions”, Int. J. Geom. Methods Mod. Phys., vol. 14, no. 7, 1750114, 25 pages, 2017.

P. Baird and J. C. Wood, Harmonic morphism between Riemannian manifolds, Oxford science publications, Oxford, 2003.

D. E. Blair, Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics. 203, Birkhäuser Boston, Basel, Berlin, 2002.

M. Cengizhan and I. K. Erken, “Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian submersions”, Filomat, vol. 29, no. 7, pp. 1429–1444, 2015.

J.-P. Bourguignon and H. B. Lawson, “Stability and isolation phenomena for Yang-Mills fields”, Comm. Math. Phys., vol. 79, no. 2, pp.189–230, 1981.

J.-P. Bourguignon, “A mathematician’s visit to Kaluza-Klein theory”, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue, pp. 143–163, 1989.

D. Chinea, “Almost contact metric submersions”, Rend. Circ. Mat. Palermo, vol. 34, no. 1, pp. 89–104, 1985.

M. Falcitelli, A. M. Pastore and S. Ianus, Riemannian submersions and related topics, World Scientific, River Edge, NJ, 2004.

A. Gray, “Pseudo-Riemannian almost product manifolds and submersions”, J. Math. Mech., vol. 16, pp. 715–738, 1967.

Y. Gündüzalp and M. A. Akyol, “Conformal slant submersions from cosymplectic manifolds”, Turkish J. Math., vol. 42, no. 5, pp. 2672–2689, 2018.

S. Ianuş and M. Visinescu, “Kaluza-Klein theory with scalar fields and generalized Hopf manifolds”, Classical Quantum Gravity, vol. 4, no. 5, pp. 1317–1325, 1987.

S. Ianuş, A. M. Ionescu, R. Mocanu and G. E. Vîlcu, “Riemannian submersions from almost contact metric manifolds”, Abh. Math. Semin. Univ. Hambg., vol. 81, no. 1, pp. 101–114, 2011.

S. Kumar, R. Prasad and P. K. Singh, “Conformal semi-slant submersions from Lorentzian para Sasakian manifolds”, Commun. Korean Math. Soc., vol. 34, no. 2, pp. 637–655, 2019.

S. Longwap, F. Massamba and N. E. Homti, “On quasi-hemi-slant Riemannian submersion”, Journal of Advances in Mathematics and Computer Science, vol. 34, no. 1, pp. 1–14, 2019.

B. O’Neill, “The fundamental equations of a submersion”, Michigan Math. J., vol. 33, no. 13, pp. 459–469, 1966.

K.-S. Park and R. Prasad, “Semi-slant submersions”, Bull. Korean Math. Soc, vol. 50, no. 3, pp. 951–962, 2013.

R. Prasad, S. S. Shukla and S. Kumar, “On quasi-bi-slant submersions”, Mediterr. J. Math., vol. 16, no. 6, paper no. 155, 18 pages, 2019.

R. Prasad, P. K. Singh and S. Kumar, “On quasi-bi-slant submersions from Sasakian manifolds onto Riemannian manifolds”, Afr. Mat., vol. 32, no. 3-4, pp. 403–417, 2020.

B. Şahin,“Semi-invariant submersions from almost Hermitian manifolds”, Canad.Math.Bull., vol. 56, no. 1, pp. 173–182, 2013.

B. Şahin, “Slant submersions from almost Hermitian manifolds”, Bull. Math. Soc. Sci. Math. Roumanie, vol. 54 (102), no. 1, pp. 93–105, 2011.

B. Şahin, “Riemannian submersion from almost Hermitian manifolds”, Taiwanese J. Math., vol. 17, no. 2, pp. 629–659, 2013.

B. Şahin, Riemannian submersions, Riemannian maps in Hermitian geometry and their applications, Elsevier, Academic Press, London, 2017.

B. Şahin, “Anti-invariant Riemannian submersions from almost Hermitian manifolds”, Cent. Eur. J. Math., vol. 8, no. 3, pp. 437–447, 2010.

C. Sayar, M. A. Akyol and R. Prasad, “Bi-slant submersions in complex geometry”, Int. J. Geom. Methods Mod. Phys., vol. 17, no. 4, 17 pages, 2020.

C. Sayar, H. M. Ta ̧stan, F. Özdemir and M. M. Tripathi, “Generic submersions from Kaehler manifolds”, Bull. Malays. Math. Sci. Soc., vol. 43, no. 1, pp. 809–831, 2020.

H. M. Taştan, B. Şahin and Ş. Yanan, “Hemi-slant submersions”, Mediterr. J. Math., vol 13, no. 4, pp. 2171–2184, 2016.

B. Watson, “Almost Hermitian submersions”, J. Differential Geometry, vol. 11, no. 1, pp. 147–165, 1976.