Czechoslovak Mathematical Journal, Vol. 70, No. 3, pp. 631-656, 2020


H-conformal anti-invariant submersions from almost quaternionic Hermitian manifolds

Kwang Soon Park

Received May 30, 2018.   Published online August 4, 2020.

Abstract:  We introduce the notions of h-conformal anti-invariant submersions and h-conformal Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, anti-invariant submersions, h-anti-invariant submersions, h-Lagrangian submersion, conformal anti-invariant submersions. We investigate their properties: the integrability of distributions, the geometry of foliations, the conditions for such maps to be totally geodesic, etc. Finally, we give some examples of such maps.
Keywords:  horizontally conformal submersion; quaternionic manifold; totally geodesic
Classification MSC:  53C15, 53C26, 53C43
DOI:  10.21136/CMJ.2020.0264-18

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] M. A. Akyol, B. Şahin: Conformal anti-invariant submersions from almost Hermitian manifolds. Turk. J. Math. 40 (2016), 43-70. DOI 10.3906/mat-1408-20 | MR 3438784 | Zbl 1424.53056
[2] D. V. Alekseevsky, S. Marchiafava: Almost complex submanifolds of quaternionic manifolds. Steps in Differential Geometry. Institute of Mathematics and Informatics, University of Debrecen, Debrecen (2001), 23-38. MR 1859285 | Zbl 1037.53029
[3] P. Baird, J. C. Wood: Harmonic Morphisms between Riemannian Manifolds. London Mathematical Society Monographs. New Series 29. Oxford University Press, Oxford (2003). DOI 10.1093/acprof:oso/9780198503620.001.0001 | MR 2044031 | Zbl 1055.53049
[4] A. L. Besse: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 10. Springer, Berlin (1987). DOI 10.1007/978-3-540-74311-8 | MR 0867684 | Zbl 0613.53001
[5] J.-P. Bourguignon: A mathematician's visit to Kaluza-Klein theory. Rend. Semin. Mat., Torino Special Issue (1989), 143-163. MR 1086213 | Zbl 0717.53062
[6] J.-P. Bourguignon, H. B. Lawson, Jr.: Stability and isolation phenomena for Yang-Mills fields. Commum. Math. Phys. 79 (1981), 189-230. DOI 10.1007/BF01942061 | MR 0612248 | Zbl 0475.53060
[7] B.-Y. Chen: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Leuven (1990). MR 1099374 | Zbl 0716.53006
[8] V. Cortés, C. Mayer, T. Mohaupt, F. Saueressig: Special geometry of Euclidean supersymmetry. I. Vector multiplets. J. High Energy Phys. 2004 (2004), Article ID 028, 73 pages. DOI 10.1088/1126-6708/2004/03/028 | MR 2061551
[9] M. Falcitelli, S. Ianus, A. M. Pastore: Riemannian Submersions and Related Topics. World Scientific, River Edge (2004). DOI 10.1142/5568 | MR 2110043 | Zbl 1067.53016
[10] B. Fuglede: Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier 28 (1978), 107-144. DOI 10.5802/aif.691 | MR 0499588 | Zbl 0339.53026
[11] A. Gray: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16 (1967), 715-737. DOI 10.1512/iumj.1967.16.16047 | MR 0205184 | Zbl 0147.21201
[12] S. Gudmundsson: The Geometry of Harmonic Morphisms. Ph.D. Thesis. University of Leeds, Leeds (1992). Available at http://www.matematik.lu.se/matematiklu/personal/sigma/Doctoral-thesis.pdf.
[13] S. Gudmundsson, J. C. Wood: Harmonic morphisms between almost Hermitian manifolds. Boll. Unione Mat. Ital., VII. Ser., B 11 (1997), 185-197. MR 1456260 | Zbl 0879.53023
[14] S. Ianuş, R. Mazzocco, G. E. Vîlcu: Riemannian submersions from quaternionic manifolds. Acta. Appl. Math. 104 (2008), 83-89. DOI 10.1007/s10440-008-9241-3 | MR 2434668 | Zbl 1151.53329
[15] S. Ianuş, M. Vişinescu: Kaluza-Klein theory with scalar fields and generalized Hopf manifolds. Classical Quantum Gravity 4 (1987), 1317-1325. DOI 10.1088/0264-9381/4/5/026 | MR 0905571 | Zbl 0629.53072
[16] S. Ianuş, M. Vişinescu: Space-time compactification and Riemannian submersions. The Mathematical Heritage of C. F. Gauss. World Scientific, River Edge (1991), 358-371. DOI 10.1142/9789814503457_0026 | MR 1146240 | Zbl 0765.53064
[17] T. Ishihara: A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ. 19 (1979), 215-229. DOI 10.1215/kjm/1250522428 | MR 0545705 | Zbl 0421.31006
[18] D. H. Jin, J. W. Lee: Conformal anti-invariant submersions from hyper-Kähler manifolds. JP J. Geom. Topol. 19 (2016), 161-183. DOI 10.17654/GT019020161 | Zbl 1358.53051
[19] M. T. Mustafa: Applications of harmonic morphisms to gravity. J. Math. Phys. 41 (2000), 6918-6929. DOI 10.1063/1.1290381 | MR 1781415 | Zbl 0974.58017
[20] B. O'Neill: The fundamental equations of a submersion. Mich. Math. J. 13 (1966), 459-469. DOI 10.1307/mmj/1028999604 | MR 0200865 | Zbl 0145.18602
[21] K.-S. Park: H-semi-invariant submersions. Taiwanese J. Math. 16 (2012), 1865-1878. DOI 10.11650/twjm/1500406802 | MR 2970690 | Zbl 1262.53028
[22] K.-S. Park: Almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds. To appear in Hacet. J. Math. Stat. DOI 10.1088/1126-6708/2004/03/028 |
[23] K.-S. Park: H-anti-invariant submersions from almost quaternionic Hermitian manifolds. Czech. Math. J. 67 (2017), 557-578. DOI 10.21136/CMJ.2017.0143-16 | MR 3661061 | Zbl 06738539
[24] K.-S. Park, R. Prasad: Semi-slant submersions. Bull. Korean Math. Soc. 50 (2013), 951-962. DOI 10.4134/BKMS.2013.50.3.951 | MR 3066240 | Zbl 1273.53023
[25] R. Ponge, H. Reckziegel: Twisted products in pseudo-Riemannian geometry. Geom. Dedicata 48 (1993), 15-25. DOI 10.1007/BF01265674 | MR 1245571 | Zbl 0792.53026
[26] B. Şahin: Anti-invariant Riemannian submersions from almost Hermitian manifolds. Cent. Eur. J. Math. 8 (2010), 437-447. DOI 10.2478/s11533-010-0023-6 | MR 2653653 | Zbl 1207.53036
[27] B. Şahin: Slant submersions from almost Hermitian manifolds. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 54 (2011), 93-105. MR 2799245 | Zbl 1224.53054
[28] B. Şahin: Riemannian submersions from almost Hermitian manifolds. Taiwanese J. Math. 17 (2013), 629-659. DOI 10.11650/tjm.17.2013.2191 | MR 3044527 | Zbl 1286.53041
[29] B. Şahin: Semi-invariant submersions from almost Hermitian manifolds. Can. Math. Bull. 56 (2013), 173-183. DOI 10.4153/CMB-2011-144-8 | MR 3009415 | Zbl 1259.53014
[30] H. Urakawa: Calculus of Variations and Harmonic Maps. Translations of Mathematical Monographs 132. American Mathematical Society, Providence (1993). DOI 10.1090/mmono/132 | MR 1252178 | Zbl 0799.58001
[31] B. Watson: Almost Hermitian submersions. J. Differ. Geom. 11 (1976), 147-165. DOI 10.4310/jdg/1214433303 | MR 0407784 | Zbl 0355.53037
[32] B. Watson: $G,G'$-Riemannian submersions and nonlinear gauge field equations of general relativity. Global Analysis - Analysis on Manifolds. Teubner Texts in Mathematics 57. Teubner, Leipzig (1983), 324-349. MR 0730623 | Zbl 0525.53052

Affiliations:   Kwang Soon Park, Division of General Mathematics, Room 4-107, Changgong Hall, University of Seoul, Seoul 02504, Republic of Korea, e-mail: parkksn@gmail.com


 
PDF available at: