Kyoto Journal of Mathematics

On some aspects of duality principle

Vladica Andrejić and Zoran Rakić

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Abstract

This paper is devoted to the study of the relation between Osserman algebraic curvature tensors and algebraic curvature tensors which satisfy the duality principle. We give a short overview of the duality principle in Osserman manifolds and extend this notion to null vectors. Here, it is proved that a Lorentzian totally Jacobi-dual curvature tensor is a real space form. Also, we find out that a Clifford curvature tensor is Jacobi-dual. We provide a few examples of Osserman manifolds which are totally Jacobi-dual and an example of an Osserman manifold which is not totally Jacobi-dual.

Article information

Source
Kyoto J. Math., Volume 55, Number 3 (2015), 567-577.

Dates
Received: 11 October 2013
Revised: 5 June 2014
Accepted: 10 June 2014
First available in Project Euclid: 9 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1441824036

Digital Object Identifier
doi:10.1215/21562261-3089064

Mathematical Reviews number (MathSciNet)
MR3395978

Zentralblatt MATH identifier
06489506

Subjects
Primary: 53C50: Lorentz manifolds, manifolds with indefinite metrics
Secondary: 53B30: Lorentz metrics, indefinite metrics 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords
Duality principle Osserman manifold Clifford structure

Citation

Andrejić, Vladica; Rakić, Zoran. On some aspects of duality principle. Kyoto J. Math. 55 (2015), no. 3, 567--577. doi:10.1215/21562261-3089064. https://projecteuclid.org/euclid.kjm/1441824036


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References

  • [1] V. Andrejić, “On certain classes of algebraic curvature tensors” in Proceedings of the 5th Summer School in Modern Mathematical Physics (Belgrade, 2008), Inst. Phys., Belgrade, 2009, 43–50.
  • [2] V. Andrejić, Strong duality principle for four-dimensional Osserman manifolds, Kragujevac J. Math. 33 (2010), 17–28.
  • [3] V. Andrejić, Duality principle for Osserman manifolds (in Serbian), Ph.D. dissertation, Belgrade, 2010.
  • [4] V. Andrejić and Z. Rakić, On the duality principle in pseudo-Riemannian Osserman manifolds, J. Geom. Phys. 57 (2007), 2158–2166.
  • [5] N. Blažić, N. Bokan, and P. Gilkey, A note on Osserman Lorentzian manifolds, Bull. Lond. Math. Soc. 29 (1997), 227–230.
  • [6] N. Blažić, N. Bokan, and Z. Rakić, Osserman pseudo-Riemannian manifolds of signature $(2,2)$, J. Aust. Math. Soc. 71 (2001), 367–395.
  • [7] M. Brozos-Vázquez, E. Garcia-Rio, P. Gilkey, S. Nikčević, and R. Vazquez-Lorenzo, The Geometry of Walker Manifolds, Synth. Lect. Math. Stat. 5, Morgan & Claypool, Williston, Vt., 2009.
  • [8] M. Brozos-Vázquez and E. Merino, Equivalence between the Osserman condition and the Rakić duality principle in dimension 4, J. Geom. Phys. 62 (2012), 2346–2352.
  • [9] Q.-S. Chi, A curvature characterization of certain locally rank-one symmetric spaces, J. Differential Geom. 28 (1988), 187–202.
  • [10] P. M. Gadea and A. Montesinos Amilibia, Spaces of constant para-holomorphic sectional curvature, Pacific J. Math. 136 (1989), 85–101.
  • [11] E. Garcia-Rio, D. N. Kupeli, and R. Vázquez-Lorenzo, Osserman Manifolds in Semi-Riemannian Geometry, Lecture Notes in Math. 1777, Springer, Berlin, 2002.
  • [12] P. B. Gilkey, Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor, World Scientific, River Edge, N.J., 2001.
  • [13] P. Gilkey, The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds, ICP Adv. Texts Math. 2, Imperial College Press, London, 2007.
  • [14] P. B. Gilkey and R. Ivanova, “Spacelike Jordan Osserman algebraic curvature tensors in the higher signature setting” in Differential Geometry, Valencia, 2001, World Scientific, River Edge, N.J., 2002, 179–186.
  • [15] Y. Nikolayevsky, Osserman manifolds of dimension 8, Manuscripta Math. 115 (2004), 31–53.
  • [16] Y. Nikolayevsky, Osserman conjecture in dimension $n\neq8,16$, Math. Ann. 331 (2005), 505–522.
  • [17] Y. Nikolayevsky, “On Osserman manifolds of dimension 16” in Contemporary Geometry and Related Topics, Univ. Belgrade Fac. Math., Belgrade, 2006, 379–398.
  • [18] Y. Nikolayevsky and Z. Rakić, A note on Rakić duality principle for Osserman manifolds, Publ. Inst. Math. (Beograd) (N.S.) 94 (2013), 43–45.
  • [19] R. Osserman, Curvature in the eighties, Amer. Math. Monthly 97 (1990), 731–756.
  • [20] Z. Rakić, On duality principle in Osserman manifolds, Linear Algebra Appl. 296 (1999), 183–189.