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Conformal slant submersions

Year 2019, Volume: 48 Issue: 1, 28 - 44, 01.02.2019

Abstract

As a generalization of conformal holomorphic submersions and conformal anti-invariant submersions, we introduce a new  conformal  submersion from almost Hermitian manifolds onto Riemannian manifolds, namely conformal slant submersions. We give examples and find necessary and sufficient conditions for such maps to be harmonic morphism. We also  investigate the geometry of foliations which are arisen from the definition of a conformal submersion and obtain a decomposition theorem on the total space of a conformal slant submersion. Moreover, we  find necessary and sufficient conditions of a conformal slant submersion to be totally geodesic.

References

  • M.A. Akyol,Conformal anti-invariant submersions from cosymplectic manifolds, Hacet. J. Math. Stat. 46 (2), 177-192, 2017.
  • M.A. Akyol and B. Sahin, Conformal anti-invariant submersions from almost Hermitian manifolds, Turk J. Math. 40 (1), 43-70, 2016.
  • C. Altafini, Redundant robotic chains on Riemannian submersions, 20, IEEE Transactions on Robotics and Automation, 335-340, 2004.
  • S. Ali and T. Fatima, Anti-invariant Riemannian submersions from nearly Kaehler manifolds, Filomat, 27, 1219-1235, 2013.
  • S. Ali and T. Fatima, Generic Riemannian submersions, Tamkang J. Math. 44, 395- 409, 2013.
  • P.Baird and J.C. Wood, Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, 29, Oxford University Press, The Clarendon Press. Oxford, 2003.
  • A. Beri, I.K. Erken and C. Murathan, Anti-Invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds, Turk. J. Math. 40, 540-552, 2016.
  • A.L. Besse, Einstein Manifolds, Springer, Berlin, 1987.
  • R. Bhattacharyaa and V. Patrangenarub, Nonparametic estimation of location and dispersion on Riemannian manifolds, J. Statist. Plann. Inference 108, 23-35, 2002.
  • J.P. Bourguingnon and H.B. Lawson, A mathematician’s visit to Kaluza-Klein theory, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue, 143-163, 1989.
  • J.L. Cabrerizo, A. Carriazo, L.M. Fernandez and M. Fernandez, Slant submanifolds in Sasakian manifolds. Glasg. Math. J. 42 (1), 125-138, 2000.
  • B.Y. Chen, Pseudo-Riemannian geometry, $\delta$-invariants and applications, World Scientific, Singapore, 2011.
  • B.Y. Chen, Slant immersions, Bull. Austral. Math. Soc. 41 (1), 135-147, 1990.
  • B.Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Leuven, 1990.
  • D. Chinea, Harmonicity on maps between almost contact metric manifolds, Acta Math. Hungar. 126, 352-365, 2010.
  • D. Chinea, Harmonicity of holomorphic maps between almost Hermitian manifolds, Canad. Math. Bull. 52, 18-27, 2009.
  • D. Chinea, D. On horizontally conformal $(\varphi,\acute{\varphi})$-holomorphic submersions, Houston J. Math. 34, 721-737, 2008.
  • I.K. Erken and C. Murathan, On slant Riemannian submersions for cosymplectic manifolds, Bull. Korean Math. Soc. 51 (6), 1749-1771, 2014.
  • I.K. Erken and C. Murathan, Anti-Invariant Riemannian submersions from Cosymplectic manifolds onto Riemannian manifolds, Filomat 29, 1429-1444, 2015.
  • I.K. Erken and C. Murathan, Slant Riemannian submersions from Sasakian manifolds, Arab J. Math. Sci. 22, 250-264, 2016.
  • M. Falcitelli, S. Ianus and A.M. Pastore, Riemannian submersions and Related Topics, World Scientific, River Edge, NJ, 2004.
  • M. Falcitelli, S. Ianus, A.M. Pastore and M. Visinescu, Some applications of Riemannian submersions in physics, Revue Roumaine de Physique 48, 627-639, 2003.
  • B. Fuglede, Harmonic Morphisms Between Riemannian Manifolds, Ann. Inst. Fourier (Grenoble) 28, 107-144, 1978.
  • P. Gilkey, M. Itoh and J.H. Park, Anti-invariant Riemannian submersions: A Lietheoretical approach, Taiwanese J. Math. 20 (4), 787-800, 2016.
  • A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16, 715-737, 1967.
  • S. Gundmundsson, The Geometry of Harmonic Morphisms, Ph.D. Thesis, University of Leeds, 1992.
  • S. Gundmundsson and J.C. Wood, Harmonic Morphisms between almost Hermitian manifolds, Boll. Un. Mat. Ital. B. 11 (2), 185-197, 1997.
  • Y. Gündüzalp, Slant submersions from almost product Riemannian manifolds, Turk. J. Math. 37 (5), 863-873, 2013.
  • Y. Gündüzalp, Anti-invariant semi-Riemannian submersions from almost para- Hermitian manifolds, J. Funct. Spaces Appl 2013, Article ID 720623, 7 pages, 2013
  • Y. Gündüzalp, Anti-invariant Riemannian submersions from almost product manifolds, Mathematical Sciences and Applications E-Notes (MSAEN) 1, 58-66, 2013.
  • Y. Gündüzalp, Slant submersions from Lorentzian almost paracontact manifolds, Gulf J. Math. 3, 18-28, 2015.
  • Y. Gündüzalp, Slant submersions from almost paracontact Riemannian manifolds, Kuwait J. Sci. 42, 17-29, 2015.
  • T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. kyoto Univ. 19, 215-229, 1979.
  • J.W. Lee, Anti-invariant $\xi^{\perp}$ Riemannian submersions from almost contact manifolds, Hacet. J. Math. Stat. 42 (3), 231-241, 2013.
  • J.C. Lee, J.H. Park, B. Sahin and D.Y. Song, Einstein conditions for the base space of anti-invariant Riemannian submersions and Clairaut Submersions, Taiwanese J. Math. 19, 1145-1160, 2015.
  • F. Memoli, G. Sapiro and P. Thompson, Implicit brain imaging, NeuroImage 23, 179-183, 2004.
  • B. O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13, 458-469, 1966.
  • K.S. Park and R. Prasad, Semi-slant submersions, Bull. Korean Math. Soc. 50 (3), 951-962, 2013.
  • R. Ponge and H. Reckziegel, Twisted products in pseudo-Riemannian geometry, Geom. Dedicata. 48 (1), 15-25, 1993.
  • S.A. Sepet and M. Ergüt, Pointwise slant submersions from cosymplectic manifolds, Turk. J. Math. 40, 582-593, 2016.
  • B. Sahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds, Central European J. Math 3, 437-447, 2010.
  • B. Sahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie 1, 93-105, 2011.
  • A.D. Vilcu and G.E. Vilcu, Statistical manifolds with almost quaternionic Structures and quaternionic Kähler-like statistical submersions, Entropy 17, 6213-6228, 2015.
  • B. Watson, B. Almost Hermitian submersions. J. Differential Geom. 11 (1), 147-165, 1976.
  • K. Yano and M. Kon, Structures on Manifolds, World Scientific, Singapore, 1984.
  • H. Zhao, A.R. Kelly, J. Zhou, L. Lu and Y.Y. Yang, Graph attribute embedding via Riemannian submersion learning, Comput. Vis. Image Underst. 115, 962-975, 2011.
Year 2019, Volume: 48 Issue: 1, 28 - 44, 01.02.2019

Abstract

References

  • M.A. Akyol,Conformal anti-invariant submersions from cosymplectic manifolds, Hacet. J. Math. Stat. 46 (2), 177-192, 2017.
  • M.A. Akyol and B. Sahin, Conformal anti-invariant submersions from almost Hermitian manifolds, Turk J. Math. 40 (1), 43-70, 2016.
  • C. Altafini, Redundant robotic chains on Riemannian submersions, 20, IEEE Transactions on Robotics and Automation, 335-340, 2004.
  • S. Ali and T. Fatima, Anti-invariant Riemannian submersions from nearly Kaehler manifolds, Filomat, 27, 1219-1235, 2013.
  • S. Ali and T. Fatima, Generic Riemannian submersions, Tamkang J. Math. 44, 395- 409, 2013.
  • P.Baird and J.C. Wood, Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, 29, Oxford University Press, The Clarendon Press. Oxford, 2003.
  • A. Beri, I.K. Erken and C. Murathan, Anti-Invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds, Turk. J. Math. 40, 540-552, 2016.
  • A.L. Besse, Einstein Manifolds, Springer, Berlin, 1987.
  • R. Bhattacharyaa and V. Patrangenarub, Nonparametic estimation of location and dispersion on Riemannian manifolds, J. Statist. Plann. Inference 108, 23-35, 2002.
  • J.P. Bourguingnon and H.B. Lawson, A mathematician’s visit to Kaluza-Klein theory, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue, 143-163, 1989.
  • J.L. Cabrerizo, A. Carriazo, L.M. Fernandez and M. Fernandez, Slant submanifolds in Sasakian manifolds. Glasg. Math. J. 42 (1), 125-138, 2000.
  • B.Y. Chen, Pseudo-Riemannian geometry, $\delta$-invariants and applications, World Scientific, Singapore, 2011.
  • B.Y. Chen, Slant immersions, Bull. Austral. Math. Soc. 41 (1), 135-147, 1990.
  • B.Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Leuven, 1990.
  • D. Chinea, Harmonicity on maps between almost contact metric manifolds, Acta Math. Hungar. 126, 352-365, 2010.
  • D. Chinea, Harmonicity of holomorphic maps between almost Hermitian manifolds, Canad. Math. Bull. 52, 18-27, 2009.
  • D. Chinea, D. On horizontally conformal $(\varphi,\acute{\varphi})$-holomorphic submersions, Houston J. Math. 34, 721-737, 2008.
  • I.K. Erken and C. Murathan, On slant Riemannian submersions for cosymplectic manifolds, Bull. Korean Math. Soc. 51 (6), 1749-1771, 2014.
  • I.K. Erken and C. Murathan, Anti-Invariant Riemannian submersions from Cosymplectic manifolds onto Riemannian manifolds, Filomat 29, 1429-1444, 2015.
  • I.K. Erken and C. Murathan, Slant Riemannian submersions from Sasakian manifolds, Arab J. Math. Sci. 22, 250-264, 2016.
  • M. Falcitelli, S. Ianus and A.M. Pastore, Riemannian submersions and Related Topics, World Scientific, River Edge, NJ, 2004.
  • M. Falcitelli, S. Ianus, A.M. Pastore and M. Visinescu, Some applications of Riemannian submersions in physics, Revue Roumaine de Physique 48, 627-639, 2003.
  • B. Fuglede, Harmonic Morphisms Between Riemannian Manifolds, Ann. Inst. Fourier (Grenoble) 28, 107-144, 1978.
  • P. Gilkey, M. Itoh and J.H. Park, Anti-invariant Riemannian submersions: A Lietheoretical approach, Taiwanese J. Math. 20 (4), 787-800, 2016.
  • A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16, 715-737, 1967.
  • S. Gundmundsson, The Geometry of Harmonic Morphisms, Ph.D. Thesis, University of Leeds, 1992.
  • S. Gundmundsson and J.C. Wood, Harmonic Morphisms between almost Hermitian manifolds, Boll. Un. Mat. Ital. B. 11 (2), 185-197, 1997.
  • Y. Gündüzalp, Slant submersions from almost product Riemannian manifolds, Turk. J. Math. 37 (5), 863-873, 2013.
  • Y. Gündüzalp, Anti-invariant semi-Riemannian submersions from almost para- Hermitian manifolds, J. Funct. Spaces Appl 2013, Article ID 720623, 7 pages, 2013
  • Y. Gündüzalp, Anti-invariant Riemannian submersions from almost product manifolds, Mathematical Sciences and Applications E-Notes (MSAEN) 1, 58-66, 2013.
  • Y. Gündüzalp, Slant submersions from Lorentzian almost paracontact manifolds, Gulf J. Math. 3, 18-28, 2015.
  • Y. Gündüzalp, Slant submersions from almost paracontact Riemannian manifolds, Kuwait J. Sci. 42, 17-29, 2015.
  • T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. kyoto Univ. 19, 215-229, 1979.
  • J.W. Lee, Anti-invariant $\xi^{\perp}$ Riemannian submersions from almost contact manifolds, Hacet. J. Math. Stat. 42 (3), 231-241, 2013.
  • J.C. Lee, J.H. Park, B. Sahin and D.Y. Song, Einstein conditions for the base space of anti-invariant Riemannian submersions and Clairaut Submersions, Taiwanese J. Math. 19, 1145-1160, 2015.
  • F. Memoli, G. Sapiro and P. Thompson, Implicit brain imaging, NeuroImage 23, 179-183, 2004.
  • B. O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13, 458-469, 1966.
  • K.S. Park and R. Prasad, Semi-slant submersions, Bull. Korean Math. Soc. 50 (3), 951-962, 2013.
  • R. Ponge and H. Reckziegel, Twisted products in pseudo-Riemannian geometry, Geom. Dedicata. 48 (1), 15-25, 1993.
  • S.A. Sepet and M. Ergüt, Pointwise slant submersions from cosymplectic manifolds, Turk. J. Math. 40, 582-593, 2016.
  • B. Sahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds, Central European J. Math 3, 437-447, 2010.
  • B. Sahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie 1, 93-105, 2011.
  • A.D. Vilcu and G.E. Vilcu, Statistical manifolds with almost quaternionic Structures and quaternionic Kähler-like statistical submersions, Entropy 17, 6213-6228, 2015.
  • B. Watson, B. Almost Hermitian submersions. J. Differential Geom. 11 (1), 147-165, 1976.
  • K. Yano and M. Kon, Structures on Manifolds, World Scientific, Singapore, 1984.
  • H. Zhao, A.R. Kelly, J. Zhou, L. Lu and Y.Y. Yang, Graph attribute embedding via Riemannian submersion learning, Comput. Vis. Image Underst. 115, 962-975, 2011.
There are 46 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Mehmet Akif Akyol

Bayram Şahin

Publication Date February 1, 2019
Published in Issue Year 2019 Volume: 48 Issue: 1

Cite

APA Akyol, M. A., & Şahin, B. (2019). Conformal slant submersions. Hacettepe Journal of Mathematics and Statistics, 48(1), 28-44.
AMA Akyol MA, Şahin B. Conformal slant submersions. Hacettepe Journal of Mathematics and Statistics. February 2019;48(1):28-44.
Chicago Akyol, Mehmet Akif, and Bayram Şahin. “Conformal Slant Submersions”. Hacettepe Journal of Mathematics and Statistics 48, no. 1 (February 2019): 28-44.
EndNote Akyol MA, Şahin B (February 1, 2019) Conformal slant submersions. Hacettepe Journal of Mathematics and Statistics 48 1 28–44.
IEEE M. A. Akyol and B. Şahin, “Conformal slant submersions”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 1, pp. 28–44, 2019.
ISNAD Akyol, Mehmet Akif - Şahin, Bayram. “Conformal Slant Submersions”. Hacettepe Journal of Mathematics and Statistics 48/1 (February 2019), 28-44.
JAMA Akyol MA, Şahin B. Conformal slant submersions. Hacettepe Journal of Mathematics and Statistics. 2019;48:28–44.
MLA Akyol, Mehmet Akif and Bayram Şahin. “Conformal Slant Submersions”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 1, 2019, pp. 28-44.
Vancouver Akyol MA, Şahin B. Conformal slant submersions. Hacettepe Journal of Mathematics and Statistics. 2019;48(1):28-44.