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Grassmannians and Conformal Structure on Absolutes

  • Sasha Anan’in
  • Eduardo C. Bento Gonçalves
  • Carlos H. GrossiEmail author
Article
  • 49 Downloads
Part of the following topical collections:
  1. Homage to Prof. W.A. Rodrigues Jr

Abstract

We study grassmannians associated with a linear space with a nondegenerate hermitian form. The geometry of these grassmannians allows us to explain the relation between a (pseudo-)riemannian projective geometry and the conformal structure on its ideal boundary (absolute). Such relation encompasses, for instance, the usual conformal structure on the absolute of real hyperbolic space, the usual conformal structure on the absolute of de Sitter space, the conformal contact structure on the absolute of complex hyperbolic space, and the causal structure on the absolute of anti-de Sitter space.

Mathematics Subject Classication

53A20 (53A35, 51M10) 

Notes

Acknowledgements

It was with great sadness that we heard of the passing of Professor Waldyr Rodrigues Jr. He was a dear friend and an exceptional scientist with whom we had the pleasure and the privilege to discuss many subjects related to mathematics and physics. Waldyr was very fond of natural and coordinate-free methods in geometry, a point of view that gave rise to this paper. We are very grateful to Alexei L. Gorodentsev and Nikolay A. Tyurin for their stimulating interest in this work and to the referees whose suggestions have greatly contributed to the exposition. This paper was partially developed while the first author was enjoying the hospitality of the IHES and, the third author, that of the MPIM.

References

  1. 1.
    Anan’in, S., Grossi, C.H., Gusevskii, N.: Complex hyperbolic structures on disc bundles over surfaces. Int. Math. Res. Notices 19, 4295–4375 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anan’in, S., Grossi, C.H.: Coordinate-free classic geometries. Moscow Math. J. 11, 633–655 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anan’in, S., Grossi, C.H.: Differential geometry of grassmannians and Plücker map. Cent. Eur. J. Math. 3, 873–884 (2012)CrossRefGoogle Scholar
  4. 4.
    Calegari, D.: Causal Geometry, Geometry and the Imagination (Blog Post at Lamington). http://wordpress.com/2009/12/10/causal-geometry (2009)
  5. 5.
    Carlip, S.: The (2 + 1)-dimensional black hole. Class. Quant. Gravity 12, 2853–2880 (1995)ADSCrossRefGoogle Scholar
  6. 6.
    Eliashberg, Y., Polterovich, L.: Partially ordered groups and geometry of contact transformations. Geometr. Funct. Anal. 10, 1448–1476 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Holst, S., Peldán, P.: Black holes and causal structure in anti-de Sitter isometric spacetimes. Class. Quant. Gravity 14, 3433–3452 (1997)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departamento de Matemática, ICMCUniversidade de São PauloSão CarlosBrazil
  2. 2.CampinasBrazil

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