Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T03:56:52.763Z Has data issue: false hasContentIssue false
  • English
  • Français

Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor

Published online by Cambridge University Press:  20 November 2018

E. Calviño-Louzao ,
E. García-Río  and
R. Vázquez-Lorenzo
E. Calviño-Louzao*
Affiliation:
Department of Geometry and Topology, Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain
E. García-Río*
Affiliation:
Department of Geometry and Topology, Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain
R. Vázquez-Lorenzo*
Affiliation:
Department of Geometry and Topology, Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain
*
estebcl@edu.xunta.es
eduardo.garcia.rio@usc.es
ravazlor@edu.xunta.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Correspondence between torsion-free connections with nilpotent skew-symmetric curvature operator and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces.

Keywords

53B3053C50Walker metricRiemann extensioncurvature operatorprojectively flat and recurrent affine connection
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 5 , 01 October 2010 , pp. 1037 - 1057
Copyright
Copyright © Canadian Mathematical Society 2010

Footnotes

Supported by projects MTM2006-01432 and PGIDIT06PXIB207054PR (Spain).

References

[1] Afifi, Z., Riemann extensions of affine connected spaces. Quart. J. Math., Oxford Ser. (2) 5(1954), 312–320. doi:10.1093/qmath/5.1.312Google Scholar
[2] Arias-Marco, T. and Kowalski, O., Classification of locally homogeneous affine connections with arbitrary torsion on 2-dimensional manifolds. Monatsh. Math. 153(2008), no. 1, 1–18. doi:10.1007/s00605-007-0494-0Google Scholar
[3] Calvi˜no-Louzao, E., Garćıa Rıo, E., and Vázquez Lorenzo, R., Four-dimensional Osserman–Ivanov–Petrova metrics of neutral signature. Classical Quantum Gravity 24(2007), no. 9, 2343–2355. doi:10.1088/0264-9381/24/9/012Google Scholar
[4] Derdzinski, A., Connections with skew-symmetric Ricci tensor on surfaces. Results Math. 52(2008), no. 3–4, 223–245. doi:10.1007/s00025-008-0307-3Google Scholar
[5] Dıaz Ramos, J. C., Garćıa-Rıo, E., and Vázquez-Lorenzo, R., Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators. J. Geom. Anal. 16(2006), no. 1, 39–52.Google Scholar
[6] Dryuma, V., The Riemann extensions in theory of differential equations and their applications. Mat. Fiz. Anal. Geom. 10(2003), no. 3, 307–325.Google Scholar
[7] Fiedler, B. and Gilkey, P., Nilpotent Szabó, Osserman and Ivanov-Petrova pseudo-Riemannian manifolds. In: Recent advances in Riemannian and Lorentzian geometries (Baltimore, MD, 2003), Contemp. Math., 337, American Mathematical Society, Providence, RI, 2003, pp. 53–63.Google Scholar
[8] Garćıa-Rıo, E., Badali, A. H., and Vázquez-Lorenzo, R., Lorentzian three-manifolds with special curvature operators. Classical Quantum Gravity 25(2008), no. 1, 015003 (13pp). doi:10.1088/0264-9381/25/1/015003Google Scholar
[9] Garćıa-Rıo, E., Kupeli, D. N., Vázquez-Abal, M. E., and Vázquez-Lorenzo, R., Affine Osserman connections and their Riemann extensions. Differential Geom. Appl. 11(1999), no. 2, 145–153. doi:10.1016/S0926-2245(99)00029-7Google Scholar
[10] Garćıa-Rıo, E., Kupeli, D. N., Vázquez-Lorenzo, R., Osserman manifolds in semi-Riemannian geometry. Lecture Notes in Mathematics, 1777, Springer-Verlag, Berlin, 2002.Google Scholar
[11] Gilkey, P. B., Geometric properties of natural operators defined by the Riemannian curvature tensor. World Scientific Publishing Co., Inc., River Edge, NJ, 2001.Google Scholar
[12] Gilkey, P., Leahy, J. V., and Sadofsky, H., Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues. Indiana Univ. Math. J. 48(1999), no. 2, 615–634.Google Scholar
[13] Gilkey, P., S. Nikčević, and Videv, V., Manifolds which are Ivanov-Petrova or k-Stanilov. J. Geom. 80(2004), no. 1–2, 82–94. doi:10.1007/s00022-003-1750-7Google Scholar
[14] Gilkey, P. and Zhang, T., Algebraic curvature tensors for indefinite metrics whose skew-symmetric curvature operator has constant Jordan normal form. Houston J. Math. 28(2002), no. 2, 311–328.Google Scholar
[15] Ivanov, S. and Petrova, I., Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues. Geom. Dedicata 70(1998), no. 3, 269–282. doi:10.1023/A:1005014507809Google Scholar
[16] Kobayashi, S. and Nomizu, K., Foundations of differential geometry. I. Interscience Publishers, New York-London, 1963.Google Scholar
[17] Kowalski, O., Opozda, B., Z. Vlášek, A classification of locally homogeneous affine connections with skew-symmetric Ricci tensor on 2-dimensional manifolds. Monatsh. Math. 130(2000), no. 2, 109–125. doi:10.1007/s006050070041Google Scholar
[18] Kowalski, O., A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach. Cent. Eur. J. Math. 2(2004), no. 1, 87–102. doi:10.2478/BF02475953Google Scholar
[19] Olszak, Z., On conformally recurrent manifolds. II. Riemann extensions. Tensor (N.S.) 49(1990), no. 1, 24–31.Google Scholar
[20] Opozda, B., A classification of locally homogeneous connections on 2-dimensional manifolds. Differential Geom. Appl. 21(2004), no. 2, 173–198. doi:10.1016/j.difgeo.2004.03.005Google Scholar
[21] Patterson, E. M. and Walker, A. G., Riemann extensions. Quart. J. Math. Oxford Ser. (2) 3(1952), 19–28. doi:10.1093/qmath/3.1.19Google Scholar
[22] Walker, A. G., Canonical form for a Riemannian space with a parallel field of null planes. Quart. J. Math. Oxford Ser. (2) 1(1950), 69–79. doi:10.1093/qmath/1.1.69Google Scholar
[23] Willmore, T. J., Riemann extensions and affine differential geometry. Results Math. 13(1988), no. 3–4, 403–408.Google Scholar
[24] Wong, Y. C., Two dimensional linear connexions with zero torsion and recurrent curvature. Monatsh. Math. 68(1964), 175–184. doi:10.1007/BF01307120Google Scholar
[25] Yano, K. and Ishihara, S., Tangent and cotangent bundles: differential geometry. Pure and Applied Mathematics, 16, Marcel Dekker, Inc., New York, 1973.Google Scholar
[26] Zhang, T., Manifolds with indefinite metrics whose skew-symmetric curvature operator has constant eigenvalues, Ph. D. Thesis, University of Oregon, 2000.Google Scholar