Skip to main content
SpringerLink
Log in

Projective Geometry

  • Chapter
  • First Online:
Book cover The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2043))

Abstract

Projective geometry enters the study of partial differential equations of mixed elliptic–hyperbolic type by a very indirect route, beginning with the following geometric variational problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alty, L.J.: Kleinian signature change. Class. Quantum Grav. 11, 2523–2536 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andersson, L., Moncrief, V.: Elliptic-hyperbolic systems and the Einstein equations. Ann. Henri Poincaré 4, 1–34 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Barceló, C., Liberati, S., Visser, M.: Analogue gravity. Living Rev. Rel. 8, 12 (2005)

    Google Scholar 

  4. Barceló, C., Liberati, S., Sonego, S., Visser, M.: Causal structure of acoustic spacetimes. New J. Phys. 186 (2004)

    Google Scholar 

  5. Beltrami, E.: Saggio di interpretazione della geometria non-euclidea. Giornale di Matematiche 6, 284–312 (1868)

    Google Scholar 

  6. Beltrami, E.: Teoria fondamentale degli spazii di curvatura costante. Annali Mat. Pura Appl. ser. 2. 2, 232–255 (1868)

    Google Scholar 

  7. Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Wiley, New York (1958)

    MATH  Google Scholar 

  8. Busemann, A.: Infinitesimale Kegelige Uberschallstromungenn Schriften. Deutsche Akad. Lufo. 7B, 105–122 (1943)

    MathSciNet  Google Scholar 

  9. Cheeger, J.: On the Hodge theory of Riemannian pseudomanifolds. Geometry of the Laplace operator. In: Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979, vol. 36, pp. 91–146. American Mathematical Society, Providence (1980)

    Google Scholar 

  10. Cheeger, J.: Spectral geometry of singular Riemannian spaces. J. Different. Geom. 18, 575–657 (1983)

    MATH  MathSciNet  Google Scholar 

  11. Darabi, F., Rastkar, A.: A quantum cosmology and discontinuous signature changing classical solutions. Gen. Relat. Gravit. 38, 1355–1366 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dereli, T., Tucker, R.W.: Signature dynamics in general relativity. Class. Quantum Grav. 10, 365–373 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Didenko, V.P.: On the generalized solvability of the Tricomi problem. Ukrain. Math. J. 25, 10–18 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dray, T., Ellis, G.F.R., Hellaby, C.: Note on signature change and Colombeau theory. Gen. Relat. Gravit. 33, 1041–1046 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dray, T., Ellis, G.F.R., Hellaby, C., Manogue,C.A.: Gravity and signature change. Gen. Relat. Gravit. 29, 591–597 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dray, T., Manogue,C.A., Tucker, R.W.: Particle production from signature change. Gen. Relat. Gravit. 23, 967–971 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ellis, G., Sumeruk, A., Coule, D., Hellaby, C.: Change of signature in classical relativity Class. Quantum Grav. 9, 1535–1554 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Embacher, F.: Actions for signature change. Phys. Rev. D 51, 6764–6777 (1995)

    Article  MathSciNet  Google Scholar 

  19. Fernando Barbero G., J.: From Euclidean to Lorentzian general relativity: The real way. Phys. Rev. D 54, 1492–1499 (1996)

    Google Scholar 

  20. Gaida, I., Hollman, H.R., Stewart, J.M.: Classical and quantum analysis of repulsive singularities in four-dimensional extended supergravity. Class. Quantum Grav. 16, 2231–2246 (1999)

    Article  MATH  Google Scholar 

  21. Gibbons, G.W., Ishibashi, A.: Topology and signature changes in braneworlds. Class. Quantum Grav. 21, 2919–2935 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gordon, W.: Zur Lichtfortpflanzung nach der Relativitatstheorie. Ann. Phys. Leipzig 72, 421–456 (1923)

    Article  MATH  Google Scholar 

  23. Gu, C.: On some differential equations of mixed type in n-dimensional spaces [in Russian]. Sci. Sin. 14, 1574–1580 (1965)

    MATH  Google Scholar 

  24. Gu, C.: A global study of extremal surfaces in 3-dimensional Minkowski space. In: Gu, C-H, Berger, M., Bryant, R. L. (eds.) Differential Geometry and Differential Equations, (Shanghai 1985) Lecture Notes in Mathematics, vol. 1255, pp. 26–33. Springer, Berlin (1985)

    Google Scholar 

  25. Gu, C.: The extremal surfaces in the 3-dimensional Minkowski space. Acta Math. Sin. n. s. 1, 173–180 (1985)

    Article  MATH  Google Scholar 

  26. Gu, C.: On the mixed partial differential equations in n independent variables, Journées Équat. aux dérivées partielles, pp. 1–2. (1980)

    Google Scholar 

  27. Gu, C.: On partial differential equations of mixed type in n independent variables. Commun. Pure Appl. Math. 34, 333–345 (1981)

    Article  MATH  Google Scholar 

  28. J. B. Hartle, J.B., Hawking, S.W., Wave function of the universe. Phys. Rev. D 28, 2960–2975 (1983)

    Google Scholar 

  29. Hayashi, N., Hirata, H.: Local existence in time of small solutions to the elliptic-hyperbolicn Davey-Stewartson system in the usual Sobolev space. Proc. Edinburgh Math. Soc. 40, 563–581 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hayward, S.A.: Comment on “Failure of standard conservation laws at a classical change of signature,” Phys. Rev. D 52, 7331–7332 (1995)

    Article  MathSciNet  Google Scholar 

  31. Hayward, S.A.: Signature change in general relativity. Class. Quantum Grav. 9, 1851–1862 (1992)

    Article  MathSciNet  Google Scholar 

  32. Hellaby, C., Dray, T.: Failure of standard conservation laws at a classical change of signature. Phys. Rev. D 49, 5096–5104 (1994)

    Article  MathSciNet  Google Scholar 

  33. Hellaby, C., Dray, T.: Reply comment: Comparison of approaches to classical signature change. Phys. Rev. D 52, 7333–7339 (1995)

    Article  MathSciNet  Google Scholar 

  34. Heidmann, J.: Relativistic Cosmology, An Introduction. Springer, Berlin (1980)

    Book  MATH  Google Scholar 

  35. Hellaby, C., Sumeruk, A., Ellis, G.F.R.: Classical signature change in the black hole topology. Int. J. Mod.Phys. D6, 211–238 (1997)

    MathSciNet  Google Scholar 

  36. Hua, L.K.: Geometrical theory of partial differential equations. In: Chern, S.S., Wen-tsün, W. (eds.), Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, pp. 627–654. Gordon and Breach, New York (1982)

    Google Scholar 

  37. Hunter, J.K., Tesdall, A.M.: Weak shock reflection. In: Givoli, D., Grote, M.J., Papanicolaou, G.C. (eds.) A Celebration of Mathematical Modeling: The Joseph B. Keller Anniversary Volume, pp. 93–112. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  38. Ji, X-H., Chen, D-Q.: Tricomi’s problems of non-homogeneous equation of mixed type in real projective plane. In: Chern, S.S., Wen-tsün, W. (eds.), Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, pp. 1257–1271. Gordon and Breach, New York (1982)

    Google Scholar 

  39. Katsanis, T.: Numerical techniques for the solution of symmetric positive linear differential equations. Ph.D. thesis, Case Institute of Technology (1967)

    Google Scholar 

  40. Katsanis, T.: Numerical solution of Tricomi equation using theory of symmetric positive differential equations. SIAM J. Numer. Anal. 6, 236–253 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  41. Keller, J.B., Blank, A.: Diffraction and reflection of pulses by wedges and corners. Commun. Pure Appl. Math. 4, 75–94 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  42. Klein, C.: Binary black hole spacetimes with helical Killing vector. Phys. Rev. D 70, 124026 (2004)

    Article  MathSciNet  Google Scholar 

  43. Kossowski, M., Kriele, M.: Signature type change and absolute time in general relativity. Class. Quantum Grav. 10, 1157–1164 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  44. Kossowski, M., Kriele, M.: The Einstein equation for signature type changing spacetimes. Proc. R. Soc. London Ser. A. 446, 115–126 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  45. Kriele, M., Martin, J.: Black holes, cosmological singularities and change of signature. Class. Quantum Grav. 12, 503–511 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  46. Liberati, S., Visser, M., Weinfurtner, S.: Naturalness in an emergent analogue spacetime. Phys. Rev. Lett. 96, 151301 (2006)

    Article  Google Scholar 

  47. Lupo, D., Morawetz, C.S., Payne, K.R.: On closed boundary value problems for equations of mixed elliptic-hyperbolic type. Commun. Pure Appl. Math. 60, 1319–1348 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  48. Lupo, D., Morawetz, C.S., Payne, K.R.: Erratum: “On closed boundary value problems for equations of mixed elliptic-hyperbolic type,” [Commun. Pure Appl. Math. 60, 1319–1348 (2007)]. Commun. Pure Appl. Math. 61, 594 (2008)

    Google Scholar 

  49. Lupo, D., Payne, K.R.: Critical exponents for semilinear equations of mixed elliptic-hyperbolic and degenerate types. Commun. Pure Appl. Math. 56, 403–424 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  50. Magnanini, R., Talenti, G.: Approaching a partial differential equation of mixed elliptic-hyperbolic type. In: Anikonov, Yu.E., Bukhageim, A.L., Kabanikhin, S.I., Romanov, V.G. (eds.) Ill-posed and Inverse Problems, pp. 263–276. VSP, Utrecht (2002)

    Google Scholar 

  51. Mars, M., Senovilla, J.M.M., Vera, R.: Signature change on the brane. Phys. Rev. Lett. 86, 4219–4222 (2001)

    Article  MathSciNet  Google Scholar 

  52. Mars, M., Senovilla, J.M.M., Vera, R.: Lorentzian and signature changing branes. Phys. Rev. D 76, 044029 (2007)

    Article  MathSciNet  Google Scholar 

  53. Mars, M., Senovilla, J.M.M., Vera, R.: Is the accelerated expansion evidence of a forthcoming change of signature? Phys. Rev. D 77, 027501 (2008)

    Article  Google Scholar 

  54. Morawetz, C.S.: A weak solution for a system of equations of elliptic-hyperbolic type. Commun. Pure Appl. Math. 11, 315–331 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  55. Otway, T.H.: Nonlinear Hodge maps. J. Math. Phys. 41, 5745–5766 (2000)

    MATH  MathSciNet  Google Scholar 

  56. Otway, T.H.: Hodge equations with change of type. Ann. Mat. Pura Appl. 181, 437–452 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  57. Otway, T.H.: Maps and fields with compressible density. Rend. Sem. Mat. Univ. Padova 111, 133–159 (2004)

    MATH  MathSciNet  Google Scholar 

  58. Otway, T.H.: Harmonic fields on the extended projective disc and a problem in optics. J. Math. Phys. 46, 113501 (2005)

    Article  MathSciNet  Google Scholar 

  59. Otway, T.H.: Erratum: “Harmonic fields on the extended projective disc and a problem in optics,” [J. Math. Phys. 46, 113501 (2005)]. J. Math. Phys. 48, 079901 (2007)

    Google Scholar 

  60. Otway, T.H.: Variational equations on mixed Riemannian-Lorentzian metrics. J. Geom. Phys. 58, 1043–1061 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  61. Page, D.N.: Susskind’s challenge to the Hartle-Hawking no-boundary proposal and possible resolutions. J. Cosmol. Astropart. Phys. 01 004 (2007)

    Article  MathSciNet  Google Scholar 

  62. Payne, K.R.: Singular metrics and associated conformal groups underlying differential operators of mixed and degenerate types. Ann. Mat. Pura Appl. 184, 613–625 (2006)

    Article  Google Scholar 

  63. Schönberg, M.: Classical theory of the point electron. Phys. Rev. 69, 211–224 (1946)

    Article  Google Scholar 

  64. Stewart, J.M.: Signature change, mixed problems and numerical relativity. Class. Quantum Grav. 18, 4983–4995 (2001)

    Article  MATH  Google Scholar 

  65. Stillwell, J.: Geometry of Surfaces. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  66. Stillwell, J.: Sources of Hyperbolic Geometry. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  67. Stokes, G.G.: Mathematical and Physical Papers, vol. 5, Appendix. Cambridge University Press, Cambridge (1880–1905)

    Google Scholar 

  68. Torre, C.G.: The helically reduced wave equation as a symmetric positive system. J. Math. Phys. 44, 6223–6232 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  69. Vakili, B., Jalalzadeh, S., Sepangi, H.R.: Classical and quantum spinor cosmology with signature change. J. Cosmol. Astropart. Phys. 006 (2005)

    Google Scholar 

  70. Weinfurtner, S.E.C.: Analog model for an expanding universe. Gen. Relat. Grav. 37, 1549–1554 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  71. Weinfurtner, S.E.C.: Signature-change events in emergent spacetimes with anisotropic scaling. J. Phys. Conf. Ser. 189, 012046 (2009)

    Article  Google Scholar 

  72. Weinfurtner, S., Liberati, S., Visser, M.: Analogue spacetime based on 2-component Bose-Einstein condensates. In: Quantum Analogues: From Phase Transitions to Black Holes and Cosmology. Lecture Notes in Physics vol. 718, pp. 115–163. Springer, Heidelberg (2007)

    Google Scholar 

  73. Weinfurtner, S.E.C., White, A., Visser, M.: Trans-Planckian physics and signature change events in Bose gas hydrodynamics. Phys. Rev. D 76, 124008 (2007)

    Article  Google Scholar 

  74. Xu, Z-F., Xu, Z-Y., Chen, G-L.: Some notes on the Busemann equation [in Chinese]. Adv. in Math. (Beijing) 16, 81–86 (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Yeshiva University, New York, USA

    Thomas H. Otway

Authors
  1. Thomas H. Otway
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Otway, T.H. (2012). Projective Geometry. In: The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type. Lecture Notes in Mathematics(), vol 2043. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24415-5_6

Download citation

Publish with us

Policies and ethics

Search

Navigation