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Interactions between para-quaternionic and Grassmannian geometry

Abstract

Almost para-quaternionic structures on smooth manifolds of dimension 2n are equivalent to almost Grassmannian structures of type (2, n). We remind the equivalence and exhibit some interrelations between subjects that were previously studied independently from the para-quaternionic and the Grassmannian point of view. In particular, we relate the respective normalization conditions, distinguished curves, and twistor constructions.

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Notes

  1. Instead of the prefix para-, various synonyms can be found in the literature; split- is probably the most often.

  2. Throughout the paper, we use the standard notation of favorite Lie groups; GL for general linear, PGL for projective linear, SL for special linear, SO for special orthogonal, Spin for spin. The corresponding Lie algebras will be denoted as \(\mathfrak {gl}\), \(\mathfrak {sl}\) and \(\mathfrak {so}\), respectively.

  3. Almost para-quaternionic structures appear under various names in the literature; the frequent one in older references is almost quaternionic structures of second type. There are also alternative equivalent definitions of the structure; see, e.g., [22] for more information.

  4. Be aware that the conventions in references are not always consistent; we follow the one in which the \(\beta \)-integrability corresponds to the anti-self-duality.

  5. Beside an enormous terminology related to concrete geometries, these curves have various general nicknames, e.g., Cartan’s circles, generalized geodesics or canonical curves.

  6. Comparing with the original formulation, we differ in the sign in the front of the term containing \({\text{ P }}\). This just reflects the difference in the definition of Rho tensor here, which we took from [9], and in [7].

References

  1. Akivis, M.A., Goldberg, V.V.: Conformal Differential Geometry and Its Generalizations. Wiley, New York (1996)

    Book  Google Scholar 

  2. Akivis, M.A., Goldberg, V.V.: Semiintegrable almost Grassmann structures. Differ Geom. Appl. 10(3), 257–294 (1999)

    MathSciNet  Article  Google Scholar 

  3. Alekseevsky, D.V., Cortés, V.: The twistor spaces of a para-quaternionic Kähler manifold. Osaka J. Math. 45(1), 215–251 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Alekseevsky, D.V., Medori, C., Tomassini, A.: Homogeneous para-Kähler Einstein manifolds. Russ. Math. Surv. 64(1), 1–43 (2009)

    Article  Google Scholar 

  5. Alekseevsky, D.V., Marchiafava, S.: Quaternionic structures on a manifold and subordinated structures. Ann. Mat. Pura Appl. Ser. IV 171, 205–273 (1996)

    MathSciNet  Article  Google Scholar 

  6. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A 362, 425–461 (1978)

    MathSciNet  Article  Google Scholar 

  7. Bailey, T.N., Eastwood, M.G.: Complex paraconformal manifolds—their differential geometry and twistor theory. Forum Math. 3(1), 61–103 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Čap, A.: Correspondence spaces and twistor spaces for parabolic geometries. J. Reine Angew. Math. 582, 143–182 (2005)

    MathSciNet  Article  Google Scholar 

  9. Čap, A., Slovák, J.: Parabolic Geometries. I. Background and General Theory. Mathematical Surveys and Monographs, vol. 154. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  10. Čap, A., Slovák, J., Žádník, V.: On distinguished curves in parabolic geometries. Transform. Groups 9(2), 143–166 (2004)

    MathSciNet  Article  Google Scholar 

  11. Crampin, M., Thompson, G.: Affine bundles and integrable almost tangent structures. Math. Proc. Camb. Philos. Soc. 98, 61–71 (1985)

    MathSciNet  Article  Google Scholar 

  12. David, L.: About the geometry of almost para-quaternionic manifolds. Differ. Geom. Appl. 27(5), 575–588 (2009)

    MathSciNet  Article  Google Scholar 

  13. Goncharov, A.B.: Generalized conformal structures on manifolds. Sel. Math. Sov. 6, 307–340 (1987)

    MathSciNet  MATH  Google Scholar 

  14. Grossman, D.A.: Torsion-free path geometries and integrable second order ODE systems. Sel. Math. New Ser. 6(4), 339–342 (2000)

    MathSciNet  Article  Google Scholar 

  15. Hrdina, J., Slovák, J.: Generalized planar curves and quaternionic geometry. Ann. Global Anal. Geom. 29(4), 343–354 (2006)

    MathSciNet  Article  Google Scholar 

  16. Ivanov, S., Minchev, I., Zamkovoy, S.: Twistor and reflector spaces of almost para-quaternionic manifolds. In: Cortés, V. (ed.) Handbook of Pseudo-Riemannian Geometry and Supersymmetry, pp. 477–496. European Mathematical Society, Zürich (2010)

  17. Kobayashi, E.: A remark on the Nijenhuis tensor. Pac. J. Math. 12, 963–977 (1962)

    MathSciNet  Article  Google Scholar 

  18. Machida, Y., Sato, H.: Twistor theory of manifolds with Grassmannian structures. Nagoya Math. J. 160, 17–102 (2000)

    MathSciNet  Article  Google Scholar 

  19. Mettler, T.: Reduction of \(\beta \)-integrable 2-Segre structures. Commun. Anal. Geom. 21(2), 331–353 (2013)

    MathSciNet  Article  Google Scholar 

  20. Penrose, R.: The twistor programme. Rep. Math. Phys. 12(1), 65–76 (1977)

    MathSciNet  Article  Google Scholar 

  21. Salamon, S.: Quaternionic Kähler manifolds. Invent. Math. 67(1), 143–171 (1982)

    MathSciNet  Article  Google Scholar 

  22. Yano, K., Ako, M.: Almost quaternion structures of the second kind and almost tangent structures. Kōdai Math. Semin. Rep. 25, 63–94 (1973)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

I am grateful to Dmitri V. Alekseevsky, Andreas Čap, Jan Slovák and Josef Šilhan for many helpful conversations.

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Correspondence to Vojtěch Žádník.

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This work was supported by the Czech Science Foundation (GAČR) under the Grant GA17-01171S.

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Žádník, V. Interactions between para-quaternionic and Grassmannian geometry. Ann Glob Anal Geom 57, 321–347 (2020). https://doi.org/10.1007/s10455-020-09701-0

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Keywords

  • Almost para-quaternionic structures
  • Almost Grassmannian structures
  • Cartan connections

Mathematics Subject Classification

  • 53C15
  • 53C05