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Nonholonomic Ricci Flows, Exact Solutions in Gravity, and Symmetric and Nonsymmetric Metrics

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Abstract

We provide a proof that nonholonomically constrained Ricci flows of (pseudo) Riemannian metrics positively result into nonsymmetric metrics (as explicit examples, we consider flows of some physically valuable exact solutions in general relativity). There are constructed and analyzed three classes of solutions of Ricci flow evolution equations defining nonholonomic deformations of Taub NUT, Schwarzschild, solitonic and pp-wave symmetric metrics into nonsymmetric ones.

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References

  1. Hamilton, R.S.: Three manifolds of positive Ricci curvature. J. Diff. Geom. 17, 255–306 (1982)

    MATH  MathSciNet  Google Scholar 

  2. Hamilton, R.S.: The formation of singularities in the Ricci flow. Surveys in Differential Geometry, vol. 2, pp. 7–136. International Press, Somerville (1995)

    Google Scholar 

  3. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 (2002)

  4. Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math.DG/0309109 (2003)

  5. Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245 (2003)

  6. Cao, H.-D., Zhu, X.-P.: Hamilton–Perelman’s proof of the Poincare conjecture and the geometrization conjecture. Asian J. Math. 10, 165–495 (2006). arXiv:math.DG/0612069

    MATH  MathSciNet  Google Scholar 

  7. Cao, H.-D., Chow, B., Chu, S.-C., Yau, S.-T. (eds.): Collected Papers on Ricci Flow. International Press, Somerville (2003)

    MATH  Google Scholar 

  8. Kleiner, B., Lott, J.: Notes on Perelman’s papers. arXiv:math.DG/0605667 (2006)

  9. Morgan, J.W., Tian, G.: Ricci flow and the Poincare conjecture. arXiv:math.DG/0607607 (2006)

  10. Vacaru, S.: Nonholonomic Ricci flows: I. Riemann metrics and Lagrange–Finsler geometry. arXiv:math.dg/0612162 (2006)

  11. Vacaru, S.: Nonholonomic Ricci flows: II. Evolution equations and dynamics. J. Math. Phys. 49, 043504 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  12. Vacaru, S.: Finsler and Lagrange geometries in Einstein and string gravity. Int. J. Geom. Methods. Mod. Phys. (IJGMMP) 5, 473–511 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Miron, R., Anastasiei, M.: Vector Bundles and Lagrange Spaces with Applications to Relativity. Geometry Balkan Press, Bucharest (1997). Translation from Romanian of Editura Academiei Romane (1987)

    MATH  Google Scholar 

  14. Miron, R., Anastasiei, M.: The Geometry of Lagrange Spaces: Theory and Applications, FTPH no. 59. Kluwer Academic, Dordrecht, Boston, London (1994)

    Google Scholar 

  15. Moffat, J.W.: New theory of gravity. Phys. Rev. D 19, 3554–3558 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  16. Moffat, J.W.: Nonsymmetric gravitational theory. Phys. Lett. B 355, 447–452 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Einstein, A.: Einheitliche Fieldtheorie von Gravitation and Electrizidät, Sitzungsberichte der Preussischen Akademie Wissebsgaften, pp. 414–419. Mathematischn-Naturwissenschaftliche Klasse (1925) translated in English by A. Unzicker and T. Case, Unified Field Theory of Gravitation and Electricity, session report from July 25, 1925, pp. 214–419. ArXiv:physics/0503046 and http://www.lrz-muenchen.de/aunzicker/ae1930.html

  18. Einstein, A.: A generalization of the relativistic theory of gravitation. Ann. of Math. 46, 578–584 (1945)

    Article  MathSciNet  Google Scholar 

  19. Eisenhart, L.P.: Generalized Riemann spaces. I. Proc. Natl. Acad. USA 37, 311–314 (1951)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Eisenhart, L.P.: Generalized Riemann spaces. II. Proc. Natl. Acad. USA 38, 505–508 (1952)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Miron, R., Atanasiu, G.: Existence et arbitrariete des connexions compatibles a une structure Riemann genarilise du type presque k-horsympletique metrique. Kodai Math. J. 6, 228–237 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  22. Atanasiu, G., Hashiguchi, M., Miron, R.: Supergeneralized Finsler Spaces. Rep. Fac. Sci. Kagoshima Univ. (Math. Phys. & Chem.) 18, 19–34 (1985)

    MATH  MathSciNet  Google Scholar 

  23. Moffat, J.W.: Review of nonsymmetric gravitational theory. In: Mann, R.B., Wesson, P. (eds.) Proceedings of the Summer Institute on Gravitation, Banff Centre, Banff, Canada. World Scientific, Singapore (1991)

  24. Moffat, J.W.: Nonsymmetric gravitational theory. J. Math. Phys. 36, 3722–3232 (1995). Erratum–ibid

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Moffat, J.W.: Noncommutative quantum gravity. Phys. Lett. B 491, 345–352 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Moffat, J.W.: Late-time inhomogeneity and acceleration without dark energy. J. Cosmol. Astropart. Phys. 0605, 001 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  27. Prokopec, T., Valkenburg, W.: The cosmology of the nonsymmetric theory of gravitation. Phys. Lett. B 636, 1–4 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  28. Vacaru, S.: Ricci flows and solitonic pp-waves. Int. J. Mod. Phys. A21, 4899–4912 (2006)

    ADS  MathSciNet  Google Scholar 

  29. Vacaru, S., Visinescu, M.: Nonholonomic Ricci flows and running cosmological constant. I. 4D Taub-NUT metrics. Int. J. Mod. Phys. A 22, 1135–1160 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Vacaru, S., Visinescu, M.: Nonholonomic Ricci flows and running cosmological constant: 3D Taub-NUT metrics. Rom. Rep. Phys. 60, 218–238 (2008). arXiv:gr-qc/0609086

    Google Scholar 

  31. Vacaru, S.: Nonholonomic Ricci flows: IV. Geometric methods, exact solutions and gravity. arXiv:0705.0728 [math-ph] (2007)

  32. Vacaru, S.: Nonholonomic Ricci flows: V. Parametric deformations of solitonic pp-waves and Schwarzschild solutions. arXiv:0705.0729 [math-ph] (2007)

  33. Vacaru, S.: Einstein Gravity, Lagrange–Finsler Geometry and Nonsymmetric Metrics. arXiv:0806.06.3810 [gr-qc] (2008)

  34. Vacaru, S., Stavrinos, P., Gaburov, E., Gonţa, D.: Clifford and Riemann-Finsler Structures in Geometric Mechanics and Gravity, Selected Works, Differential Geometry—Dynamical Systems, Monograph 7. Geometry Balkan Press, Bucharest (2006). www.mathem.pub.ro/dgds/mono/va-t.pdf and gr-qc/0508023

    Google Scholar 

  35. Chamblin, A., Emparan, R., Johnson, C.V., Myers, R.C.: Large N Phases, gravitational instantons, and the nuts and bolts of AdS holography. Phys. Rev. D 59, 064010 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  36. Alonso-Alberca, N., Meessen, P., Ortin, T.: Supersymmetry of topological Kerr–Newumann–Taub–NUT–AdS spacetimes. Class. Quantum Gravity 17, 2783–2798 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mann, R., Stelea, C.: Nuttier (A)dS black holes in higher dimensions. Class. Quantum Gravity 21, 2937 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Peres, A.: Some gravitational waves. Phys. Rev. Lett. 3, 571–572 (1959)

    Article  MATH  ADS  Google Scholar 

  39. Heusler, M.: Black Hole Uniqueness Theorems. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  40. Vacaru, S.: Horizons and geodesics of black ellipsoids. Int. J. Mod. Phys. D 12, 479–494 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Vacaru, S.: Perturbations and stability of black ellipsoids. Int. J. Mod. Phys. D 12, 461–478 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. Vacaru, S.: Exact solutions with noncommutative symmetries in Einstein and gauge gravity. J. Math. Phys. 46, 042503 (2005)

    Article  ADS  MathSciNet  Google Scholar 

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  1. The Fields Institute for Research in Mathematical Science, 222 College Street, 2d Floor, Toronto, M5T 3J1, Canada

    Sergiu I. Vacaru

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  1. Sergiu I. Vacaru
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Vacaru, S.I. Nonholonomic Ricci Flows, Exact Solutions in Gravity, and Symmetric and Nonsymmetric Metrics. Int J Theor Phys 48, 579–606 (2009). https://doi.org/10.1007/s10773-008-9841-8

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  • DOI: https://doi.org/10.1007/s10773-008-9841-8

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