Conformal generic submersions

Akyol and Şahin (2017) introduced the notion of conformal semiinvariant submersions from almost Hermitian manifolds. The present paper deals with the study of conformal generic submersions from almost Hermitian manifolds which extends semiinvariant Riemannian submersions, generic Riemannian submersions and conformal semiinvariant submersions in a natural way. We mention some examples of such maps and obtain characterizations and investigate some properties, including the integrability of distributions, the geometry of foliations and totally geodesic foliations. Moreover, we obtain some conditions for such submersions to be totally geodesic and harmonic, respectively.

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[1] Ali S, Fatima T. Generic Riemannian submersions. Tamkang Journal of Mathematics 2013; 44 (4): 395-409.
[2] Akyol MA. Generic Riemannian submersions from almost product Riemannian manifolds. Gazi University Journal of Science 2017; 30 (3): 89-100.
[3] Akyol MA. Conformal semi-slant submersions. International Journal of Geometric Methods in Modern Physics 2017; 14 (7): 1750114.
[4] Akyol MA. Conformal semi-invariant submersions from almost product Riemannian manifolds. Acta Mathematica Vietnamica 2017; 42 (3): 491-507. doi: 10.1007/s40306-016-0193-9
[5] Akyol MA, Gündüzalp Y. Semi-invariant semi-Riemannian submersions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 2018; 67 (1): 80-92.
[6] Akyol MA, Şahin B. Conformal anti-invariant submersions from almost Hermitian manifolds. Turkish Journal of Mathematics 2016; 40 (1): 43-70.
[7] Akyol MA, Şahin B. Conformal semi-invariant submersions. Communications in Contemporary Mathematics 2017; 19 (2): 1650011. doi: 10.1142/S0219199716500115
[8] Akyol MA, Şahin B. Conformal slant submersions. Hacettepe Journal of Mathematics and Statistics 2019; 48 (1): 28-44.
[9] Baird P, Wood JC. Harmonic Morphisms Between Riemannian Manifolds. London Mathematical Society Monographs, New Series 29. Oxford, UK: Oxford University Press, 2003.
[10] Bejancu A. Geometry of CR-submanifolds. Norwell, MA, USA: Kluwer Academic, 1986.
[11] Caldarella AV. On paraquaternionic submersions between paraquaternionic Kähler manifolds. Acta Applicandae Mathematicae 2010; 112 (1): 1-14.
[12] Chen BY. Differential geometry of real submanifolds in a Kaehler manifold. Monatshefte für Mathematik 1981; 91: 257-274.
[13] Chinea D. On horizontally conformal (φ, φ ′ )-holomorphic submersions. Houston Journal of Mathematics 2008; 34 (3): 721-737.
[14] Choe YW, Ki UH, Takagi R. Compact minimal generic submanifolds with parallel normal section in a complex projective space. Osaka Journal of Mathematics 2000; 37 (2): 489-499.
[15] De UC, Sengupta AK, Calin C. Generic submanifolds of quasi-Sasakian manifolds. Demonstratio Mathematica 2004; 37 (2): 429-437.
[16] De UC, Sengupta AK. Generic submanifolds of a Lorentzian para-Sasakian manifold. Soochow Journal of Mathematics 2001; 27 (1): 29-36.
[17] Eells J, Sampson JH. Harmonic mappings of Riemannian manifolds. American Journal of Mathematics 1964; 86: 109–160.
[18] Fuglede B. Harmonic morphisms between Riemannian manifolds. Annales de l’institut Fourier 1978; 28: 107-144.
[19] Falcitelli M, Ianus S, Pastore AM. Riemannian submersions and related topics. River Edge, NJ, USA: World Scientific, 2004.
[20] Gundmundsson S, Wood JC. Harmonic morphisms between almost Hermitian manifolds. Bollettino dell’Unione Matematica Italiana 1997; 11 (2): 185-197.
[21] Gray A. Pseudo-Riemannian almost product manifolds and submersions. Journal of Applied Mathematics and Mechanics 1997; 16: 715-737.
[22] Ishihara T. A mapping of Riemannian manifolds which preserves harmonic functions. Journal of Mathematics of Kyoto University 1979; 19: 215-229.
[23] Ianus S, Mazzocco R, Vilcu GE. Riemannian submersions from quaternionic manifolds. Acta Applicandae Mathematicae 2008; 104 (1): 83-89.
[24] Kim JS, Choi J, Tripathi MM. On generic submanifolds of manifolds equipped with a hypercosymplectic 3-structure. Communications of the Korean Mathematical Society 2006; 21 (2): 321-335.
[25] Kon M. On minimal generic submanifolds immersed in S 2m+1 . Colloquium Mathematicum 2001; 90 (2): 299-304.
[26] Marrero JC, Rocha J. Locally conformal Kähler submersions. Geometriae Dedicata 1994; 52 (3): 271-289.
[27] O’Neill B. The fundamental equations of a submersion. Michigan Mathematical Journal 1966; 13: 458-469.
[28] Özdemir F, Sayar C, Taştan HM. Semi-invariant submersions whose total manifolds are locally product Riemannian. Quaestiones Mathematicae 2017; 40 (7): 909-929.
[29] Park KS. Almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds. Hacettepe Journal of Mathematics 2020; 49 (5): 1804-1824.
[30] Park KS. H-semi-invariant submersions. Taiwanese Journal of Mathematics 2012; 16 (5): 1865-1878.
[31] Park KS, Prasad R. Semi-slant submersions. Bulletin of the Korean Mathematical Society 2013; 50 (3): 951-962.
[32] Ronsse GB. Generic and skew CR-submanifolds of a Kaehler manifold. Bulletin of the Institute of Mathematics Academia Sinica 1990; 18: 127-141.
[33] Sayar C, Taştan HM, Özdemir F, Tripathi MM. Generic submersions from Kaehler manifolds. Bulletin of the Malaysian Mathematical Sciences Society 2020; 43 (2): 809-831. doi: 10.1007/s40840-018- 00716-2
[34] Şahin B. Anti-invariant Riemannian submersions from almost Hermitian manifolds. Central European Journal of Mathematics 2010; 3: 437-447.
[35] Şahin B. Semi-invariant Riemannian submersions from almost Hermitian manifolds. Canadian Mathematical Bulletin 2011; 56: 173-182.
[36] Şahin B. Slant submersions from almost Hermitian manifolds. Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie 2011; 54 (1): 93-105.
[37] Şahin B. Riemannian submersions from almost Hermitian manifolds. Taiwanese Journal of Mathematics 2013; 17 (2): 629-659.
[38] Şahin B. Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and Their Applications. Amsterdam, Netherlands: Elsevier Science Publishing Co., Inc., 2017.
[39] Vilcu GE. On generic submanifolds of manifolds endowed with metric mixed 3-structures. Communications in Contemporary Mathematics 2016; 18 (6): 1550081.
[40] Vilcu GE. Mixed paraquaternionic 3-submersions. Indagationes Mathematicae 2013; 24 (2): 474-488.
[41] Vilcu AD, Vîlcu GE. Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions. Entropy 2015; 17 (9): 6213-6228.
[42] Watson B. Almost Hermitian submersions. Journal of Differential Geometry 1976; 11 (1): 147-165.
[43] Urakawa H. Calculus of Variations and Harmonic Maps. Providence, RI, USA: American Mathematical Society, 1993.
[44] Yano K, Kon M. Generic submanifolds. Annali di Matematica Pura ed Applicata 1980; 123: 59-92.
[45] Yano K, Kon M. Structures on Manifolds. Singapore: World Scientific, 1984.