Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 557-578, 2017


H-anti-invariant submersions from almost quaternionic Hermitian manifolds

Kwang-Soon Park

Received March 23, 2016.  First published March 29, 2017.

Abstract:  As a generalization of anti-invariant Riemannian submersions and Lagrangian Riemannian submersions, we introduce the notions of h-anti-invariant submersions and h-Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations and investigate some properties: the integrability of distributions, the geometry of foliations, and the harmonicity of such maps. We also find a condition for such maps to be totally geodesic and give some examples of such maps. Finally, we obtain some types of decomposition theorems.
Keywords:  Riemannian submersion; Lagrangian Riemannian submersion; decomposition theorem; totally geodesic
Classification MSC:  53C15, 53C26
DOI:  10.21136/CMJ.2017.0143-16


References:
[1] C. Altafini: Redundant robotic chains on Riemannian submersions. IEEE Transactions on Robotics and Automation 20 (2004), 335-340. DOI 10.1109/tra.2004.824636
[2] D. V. Alekseevsky, S. Marchiafava: Almost complex submanifolds of quaternionic manifolds. Steps in differential geometry (Kozma, L. et al. eds.). Proc. of the colloquium on differential geometry, Debrecen, 2000, Inst. Math. Inform. 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, 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. Conf. on Partial Differential Equations and Geometry, Torino, 1988, Rend. Sem. Mat. Univ. Politec. 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] D. Chinea: Almost contact metric submersions. Rend. Circ. Mat. Palermo II. Ser. 34 (1985), 89-104. DOI 10.1007/BF02844887 | MR 0790818 | Zbl 0572.53033
[8] V. Cortés, C. Mayer, T. Mohaupt, F. Saueressig: Special geometry of Euclidean supersymmetry I. Vector multiplets. J. High Energy Phys. (electronic) 3 (2004), no. 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 Publishing, River Edge (2004). DOI 10.1142/5568 | MR 2110043 | Zbl 1067.53016
[10] A. Gray: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16 (1967), 715-737. MR 0205184 | Zbl 0147.21201
[11] S. Ianuş, R. Mazzocco, G. E. Vilcu: Riemannian submersions from quaternionic manifolds. Acta. Appl. Math. 104 (2008), 83-89. DOI 10.1007/s10440-008-9241-3 | MR 2434668 | Zbl 1151.53329
[12] S. Ianus, M. Vişinescu: Kaluza-Klein theory with scalar fields and generalised Hopf manifolds. Classical Quantum Gravity 4 (1987), 1317-1325. DOI 10.1088/0264-9381/4/5/026 | MR 0905571 | Zbl 0629.53072
[13] S. Ianus, M. Visinescu: Space-time compactification and Riemannian submersions. The Mathematical Heritage of C. F. Gauss, Collect. Pap. Mem. C. F. Gauss, World Sci. Publ., River Edge (1991), 358-371. DOI 10.1142/9789814503457_0026 | MR 1146240 | Zbl 0765.53064
[14] J. C. Marrero, J. Rocha: Locally conformal Kähler submersions. Geom. Dedicata 52 (1994), 271-289. DOI 10.1007/BF01278477 | MR 1299880 | Zbl 0810.53054
[15] F. Mémoli, G. Sapiro, P. Thompson: Implicit brain imaging. NeuroImage 23 (2004), 179-188. DOI 10.1016/j.neuroimage.2004.07.072
[16] 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
[17] B. O'Neill: The fundamental equations of a submersion. Mich. Math. J. 13 (1966), 458-469. DOI 10.1307/mmj/1028999604 | MR 0200865 | Zbl 0145.18602
[18] K.-S. Park: H-semi-invariant submersions. Taiwanese J. Math. 16 (2012), 1865-1878. MR 2970690 | Zbl 1262.53028
[19] K.-S. Park: H-slant submersions. Bull. Korean Math. Soc. 49 (2012), 329-338. DOI 10.4134/BKMS.2012.49.2.329 | MR 2934483 | Zbl 1237.53016
[20] K.-S. Park: H-semi-slant submersions from almost quaternionic Hermitian manifolds. Taiwanese J. Math. 18 (2014), 1909-1926. DOI 10.11650/tjm.18.2014.4079 | MR 3284038 | Zbl 06693490
[21] 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
[22] B. Ṣahin: 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
[23] 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
[24] B. Watson: Almost Hermitian submersions. J. Differ. Geom. 11 (1976), 147-165. DOI 10.4310/jdg/1214433303 | MR 0407784 | Zbl 0355.53037
[25] B. Watson: $G,G'$-Riemannian submersions and non-linear gauge field equations of general relativity. Global Analysis - Analysis on Manifolds Teubner-Texte Math. 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

Springer subscribers can access the papers on Springer website.
Access to full texts on this site is restricted to subscribers of Myris Trade. To activate your access, please send an e-mail to myris@myris.cz.
[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

 
PDF available at: