Table of Contents
Geometry
Volume 2014, Article ID 616487, 12 pages
http://dx.doi.org/10.1155/2014/616487
Research Article

Paracomplex Paracontact Pseudo-Riemannian Submersions

Department of Mathematics, University of Allahabad, Allahabad 211002, India

Received 25 February 2014; Accepted 7 April 2014; Published 7 May 2014

Academic Editor: Bennett Palmer

Copyright © 2014 S. S. Shukla and Uma Shankar Verma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. B. O'Neill, “The fundamental equations of a submersion,” The Michigan Mathematical Journal, vol. 13, pp. 459–469, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, vol. 103 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1983. View at MathSciNet
  3. A. Gray, “Pseudo-Riemannian almost product manifolds and submersions,” vol. 16, pp. 715–737, 1967. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. P. Bourguignon and H. B. Lawson, A Mathematicians Visit to Kaluza-Klein Theory, Rendiconti del Seminario Matematico, 1989.
  5. S. Ianus and M. Visinescu, “Space-time compactification and Riemannian submersions,” in The Mathematical Heritage, G. Rassias and C. F. Gauss, Eds., pp. 358–371, World Scientific, River Edge, NJ, USA, 1991. View at Google Scholar · View at Zentralblatt MATH
  6. J. P. Bourguignon and H. B. Lawson, Jr., “Stability and isolation phenomena for Yang-Mills fields,” Communications in Mathematical Physics, vol. 79, no. 2, pp. 189–230, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. B. Watson, “G, G'-Riemannian submersions and nonlinear gauge field equations of general relativity,” in Global Analysis—Analysis on Manifolds, T. Rassias and M. Morse, Eds., vol. 57 of Teubner-Texte zur Mathematik, pp. 324–349, Teubner, Leipzig, Germany, 1983. View at Google Scholar · View at MathSciNet
  8. C. Altafini, “Redundant robotic chains on Riemannian submersions,” IEEE Transactions on Robotics and Automation, vol. 20, no. 2, pp. 335–340, 2004. View at Publisher · View at Google Scholar · View at Scopus
  9. M. T. Mustafa, “Applications of harmonic morphisms to gravity,” Journal of Mathematical Physics, vol. 41, no. 10, pp. 6918–6929, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. B. Watson, “Almost Hermitian submersions,” Journal of Differential Geometry, vol. 11, no. 1, pp. 147–165, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. D. Chinea, “Almost contact metric submersions,” Rendiconti del Circolo Matematico di Palermo II, vol. 34, no. 1, pp. 319–330, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. D. Chinea, “Transference of structures on almost complex contact metric submersions,” Houston Journal of Mathematics, vol. 14, no. 1, pp. 9–22, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Y. Gündüzalp and B. Sahin, “Para-contact para-complex pseudo-Riemannian submersions,” Bulletin of the Malaysian Mathematical Sciences Society.
  14. I. Sato, “On a structure similar to the almost contact structure,” Tensor, vol. 30, no. 3, pp. 219–224, 1976. View at Google Scholar · View at Zentralblatt MATH
  15. P. K. Rashevskij, “The scalar field in a stratified space,” Trudy Seminara po Vektornomu i Tenzornomu Analizu s ikh Prilozheniyami k Geometrii, Mekhanike i Fizike, vol. 6, pp. 225–248, 1948. View at Google Scholar · View at MathSciNet
  16. P. Libermann, “Sur les structures presque paracomplexes,” Comptes Rendus de l'Académie des Sciences I, vol. 234, pp. 2517–2519, 1952. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. E. M. Patterson, “Riemann extensions which have Kähler metrics,” Proceedings of the Royal Society of Edinburgh A. Mathematics, vol. 64, pp. 113–126, 1954. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Sasaki, “On differentiable manifolds with certain structures which are closely related to almost contact structure I,” The Tohoku Mathematical Journal, vol. 12, pp. 459–476, 1960. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Zamkovoy, “Canonical connections on paracontact manifolds,” Annals of Global Analysis and Geometry, vol. 36, no. 1, pp. 37–60, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, vol. 23 of Progress in Mathematics, Birkhäuser, Boston, Mass, USA, 2002.
  21. V. Cruceanu, P. Fortuny, and P. M. Gadea, “A survey on paracomplex geometry,” The Rocky Mountain Journal of Mathematics, vol. 26, no. 1, pp. 83–115, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. P. M. Gadea and J. M. Masque, “Classification of almost para-Hermitian manifolds,” Rendiconti di Matematica e delle sue Applicazioni VII, vol. 7, no. 11, pp. 377–396, 1991. View at Google Scholar · View at MathSciNet
  23. M. Falcitelli, S. Ianus, and A. M. Pastore, Riemannian Submersions and Related Topics, World Scientific, River Edge, NJ, USA, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  24. E. G. Rio and D. N. Kupeli, Semi-Riemannian Maps and Their Applications, Kluwer Academic Publisher, Dordrecht, The Netherlands, 1999. View at MathSciNet
  25. B. Sahin, “Semi-invariant submersions from almost Hermitian manifolds,” Canadian Mathematical Bulletin, vol. 54, no. 3, 2011. View at Google Scholar