Skip to main content
SpringerLink
Log in

Curvature Homogeneous Pseudo-Riemannian Manifolds which are not Locally Homogeneous

  • Chapter
Complex, Contact and Symmetric Manifolds

Part of the book series: Progress in Mathematics ((PM,volume 234))

Summary

We construct a family of balanced signature pseudo-Riemannian manifolds, which arise as hypersurfaces in flat space, that are curvature homogeneous, that are modeled on a symmetric space, and that are not locally homogeneous.

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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. E. Boeckx? O. Kowalski, and L. Vanhecke, Riemannian manifolds of conullity two, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

    Google Scholar 

  2. E. Boeckx and L. Vanhecke, Curvature homogeneous unit tangent sphere bundles, Publ. Math. 53 (1998), 389–413.

    MATH  MathSciNet  Google Scholar 

  3. A. Bonome, P. Castro, and E. García-Río, Generalized Osserman four-dimensional manifolds, Classical Quantum Gravity 18 (2001), 4813–4822.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Bonome, R. Castro, E. García-Río, L. M. Hervella, R. Vázquez-Lorenzo, Nonsymmetric Osserman indefinite Kähler manifolds, Proc. Amer. Math. Soc., 126 (1998), 2763–2769.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Bueken and M. Djorić, Three-dimensional Lorentz metrics and curvature homogeneity of order one, Ann. Global Anal. Geom. 18 (2000), 85–103.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Bueken and L. Vanhecke, Examples of curvature homogeneous Lorentz metrics, Classical Quantum Gravity 14 (1997), L93–L96.

    Article  MathSciNet  Google Scholar 

  7. M. Cahen, J. Leroy, M. Parker, F. Tricerri, and L. Vanhecke, Lorentz manifolds modelled on a Lorentz symmetric space, J. Geom. Phys. 7 (1990), 571–581.

    Article  MATH  MathSciNet  Google Scholar 

  8. E. García-Río, D. Kupeli, and R. Vázquez-Lorenzo, Osserman Manifolds in Semi-Riemannian Geometry, Lecture notes in Mathematics, Springer Verlag, (2002), ISBN 3-540-43144-6.

    Google Scholar 

  9. P. Gilkey, Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor, World Scientific ISBN 981-02-4752-4 (2002).

    Google Scholar 

  10. P. Gilkey, R. Ivanova, and T. Zhang, Higher-order Jordan Osserman pseudo-Riemannian manifolds, Classical and Quantum Gravity, 19 (2002), 4543–4551; math.DG/0205269.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Gilkey, R. Ivanova, and T. Zhang, Szabó Osserman IP Pseudo-Riemannian manifolds, Publ. Math. Debrecen 62 (2003), 387–401; math.DG/0205085.

    MATH  MathSciNet  Google Scholar 

  12. R. Ivanova and G. Stanilov, A skew-symmetric curvature operator in Riemannian geometry, Sympos. Gaussiana, Conf A, ed. Behara, Fritsch, and Lintz (1995), 391–395.

    Google Scholar 

  13. I. Kath, Killing spinors on pseudo-Riemannian manifolds, Habilitation, Humboldt Universität zu Berlin (2000).

    Google Scholar 

  14. O. Kowalski, On curvature homogeneous spaces, Cordero, L. A. (ed.) et al., Proceedings of the workshop on recent topics in differential geometry, Santiago de Compostela, Spain, July 16–19, 1997. Santiago de Compostela: Universidade de Santiago de Compostela. Publ. Dep. Geom. Topologá, Univ. Santiago Compostela. 89, 193–205 (1998).

    Google Scholar 

  15. O. Kowalski, B. Opozda, and Z. Vlvsek, Curvature homogeneity of affine connections on two-dimensional manifolds, Colloq. Math. 81 (1999), 123–139.

    MATH  MathSciNet  Google Scholar 

  16. O. Kowalski, F. Tricerri, and L. Vanhecke, New examples of non-homogeneous Riemannian manifolds whose curvature tensor is that of a Riemannian symmetric space, C. R. Acad. Sci., Paris, Sér. I 311 (1990), 355–360.

    MATH  MathSciNet  Google Scholar 

  17. O. Kowalski, F. Tricerri and L. Vanhecke, Curvature homogeneous Riemannian manifolds, J. Math. Pures Appl., 71 (1992), 471–501.

    MATH  MathSciNet  Google Scholar 

  18. B. Opozda, On curvature homogeneous and locally homogeneous affine connections, Proc. Am. Math. Soc. 124 (1996), 1889–1893.

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Osserman, Curvature in the eighties, Amer. Math. Monthly, 97, (1990) 731–756.

    Article  MATH  MathSciNet  Google Scholar 

  20. V. Oproiu, Harmonic maps between tangent bundles, Rend. Sem. Mat. Univ. Politec. Torino 47 (1989), 47–55.

    MATH  MathSciNet  Google Scholar 

  21. G. Stanilov and V. Videv, Four dimensional pointwise Osserman manifolds, Abh. Math. Sem. Univ. Hamburg, 68, (1998), 1–6.

    Article  MATH  MathSciNet  Google Scholar 

  22. Z. I. Szabó, A short topological proof for the symmetry of 2 point homogeneous spaces, Invent. Math., 106, (1991), 61–64.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Tomassini, Curvature homogeneous metrics on principal fibre bundles, Ann. Mat. Pura Appl., IV. Ser. 172 (1997), 287–295.

    Article  MATH  MathSciNet  Google Scholar 

  24. F. Tricerri, Riemannian manifolds which have the same curvature as a homogeneous space, and a conjecture of Gromov, Riv. Mat. Univ. Parma, IV. Ser. 14 (1988), 91–104.

    MathSciNet  Google Scholar 

  25. L. Vanhecke, Curvature homogeneity and related problems, Recent topics in differential geometry, Proc. Workshop/Puerto de la Cruz/Spain 1990, Ser. Inf. 32 (1991), 103–122.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, University of Oregon, Eugene, OR, 97403, USA

    Corey Dunn & Peter B. Gilkey

Authors
  1. Corey Dunn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter B. Gilkey
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, 186 75, Praha, Czech Republic

    Oldřich Kowalski

  2. Dipartimento di Matematica Pura ed Applicata, Università di L’Aquila, 67100, L’Aquila, Italy

    Emilio Musso

  3. Dipartimento di Matematica “E. De Giorgi”, Università degli Studi di Lecce, 73100, Lecce, Italy

    Domenico Perrone

Additional information

Dedicated to Professor Lieven Vanhecke

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Boston

About this chapter

Cite this chapter

Dunn, C., Gilkey, P.B. (2005). Curvature Homogeneous Pseudo-Riemannian Manifolds which are not Locally Homogeneous. In: Kowalski, O., Musso, E., Perrone, D. (eds) Complex, Contact and Symmetric Manifolds. Progress in Mathematics, vol 234. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4424-5_10

Download citation

Publish with us

Policies and ethics

Search

Navigation