Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 005, 8 pages      arXiv:1001.1994      https://doi.org/10.3842/SIGMA.2010.005

Algebraic Properties of Curvature Operators in Lorentzian Manifolds with Large Isometry Groups

Giovanni Calvaruso a and Eduardo García-Río b
a) Dipartimento di Matematica "E. De Giorgi", Università del Salento, Lecce, Italy
b) Faculty of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain

Received October 01, 2009, in final form January 07, 2010; Published online January 12, 2010

Abstract
Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and ε-spaces exhaust the class of n-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least ½n(n−1)+1, for almost all values of n [Patrangenaru V., Geom. Dedicata 102 (2003), 25-33]. We shall prove that the curvature tensor of these spaces satisfy several interesting algebraic properties. In particular, we will show that Egorov spaces are Ivanov-Petrova manifolds, curvature-Ricci commuting (indeed, semi-symmetric) and P-spaces, and that ε-spaces are Ivanov-Petrova and curvature-curvature commuting manifolds.

Key words: Lorentzian manifolds; skew-symmetric curvature operator; Jacobi, Szabó and skew-symmetric curvature operators; commuting curvature operators; IP manifolds; C-spaces and P-spaces.

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References

  1. Batat W., Calvaruso G., De Leo B., Curvature properties of Lorentzian manifolds with large isometry groups, Math. Phys. Anal. Geom. 12 (2009), 201-217.
  2. Berndt J., Vanhecke L., Two natural generalizations of locally symmetric spaces, Differential Geom. Appl. 2 (1992), 57-80.
  3. Brozos-Vázquez M., Fiedler B., García-Río E., Gilkey P., Nikčević S., Stanilov G., Tsankov Y., Vázquez-Lorenzo R., Videv V., Stanilov-Tsankov-Videv theory, SIGMA 3 (2007), 095, 13 pages, arXiv:0708.0957.
  4. Brozos-Vázquez M., García-Río E., Gilkey P., Nikčević S., Vázquez-Lorenzo R., The geometry of Walker manifolds, Synthesis Lectures on Mathematics and Statistics, Vol. 5, Morgan & Claypool Publ., 2009.
  5. Calvaruso G., Three-dimensional Ivanov-Petrova manifolds, J. Math. Phys. 50 (2009), 063509, 12 pages.
  6. Calviño-Louzao E., García-Río E., Vázquez-Abal M.E., Vázquez-Lorenzo R., Curvature operators and generalizations of symmetric spaces in Lorentzian geometry, Preprint, 2009.
  7. Cahen M., Leroy J., Parker M., Tricerri F., Vanhecke L., Lorentz manifolds modelled on a Lorentz symmetric space, J. Geom. Phys. 7 (1990), 571-581.
  8. Egorov I.P., Riemannian spaces of the first three lacunary types in the geometric sense, Dokl. Akad. Nauk. SSSR 150 (1963), 730-732.
  9. García-Río E., Haji-Badali A., Vázquez-Lorenzo R., Lorentzian three-manifolds with special curvature operators, Classical Quantum Gravity 25 (2008), 015003, 13 pages.
  10. García-Río E., Haji-Badali A., Vázquez-Abal M.E., Vázquez-Lorenzo R., Lorentzian 3-manifolds with commuting curvature operators, Int. J. Geom. Methods Mod. Phys. 5 (2008), 557-572.
  11. Gilkey P., Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues. II, in Differential Geometry and Applications (Brno, 1998), Masaryk Univ., Brno, 1999, 73-87.
  12. Gilkey P., Geometric properties of natural operators defined by the Riemann curvature tensor, World Scientific Publishing Co. Inc., River Edge, NJ, 2001.
  13. Gilkey P., Leahy J.V., Sadofsky H., Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues, Indiana Univ. Math. J. 48 (1999), 615-634.
  14. Gilkey P., Nikčević S., Pseudo-Riemannian Jacobi-Videv manifolds, Int. J. Geom. Methods Mod. Phys. 4 (2007), 727-738, arXiv:0708.1096.
  15. Gilkey P., Nikčević S., The classification of simple Jacobi-Ricci commuting algebraic curvature tensors, Note Mat. 28 (2008), suppl. 1, 341-348, arXiv:0710.2080.
  16. Gilkey P., Zhang T., Algebraic curvature tensors for indefinite metrics whose skew-symmetric curvature operator has constant Jordan normal form, Houston J. Math. 28 (2002), 311-328, math.DG/0205079.
  17. Haesen S., Verstraelen L., Natural intrinsic geometrical symmetries, SIGMA 5 (2009), 086, 14 pages, arXiv:0909.0478.
  18. Hall G.S., Symmetries and curvature structure in general relativity, World Scientific Lecture Notes in Physics, Vol. 46, World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
  19. Ivanov S., Petrova I., Riemannian manifolds in which certain curvature operator has constant eigenvalues along each circle, Ann. Global Anal. Geom. 15 (1997), 157-171.
  20. Nikolayevsky Y., Riemannian manifolds whose curvature operator R(X,Y) has constant eigenvalues, Bull. Austral. Math. Soc. 70 (2004), 301-319, math.DG/0311429.
  21. Patrangenaru V., Lorentz manifolds with the three largest degrees of symmetry, Geom. Dedicata 102 (2003), 25-33.
  22. Patrangenaru V., Locally homogeneous pseudo-Riemannian manifolds, J. Geom. Phys. 17 (1995), 59-72.


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