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

 
PDF available at: