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Siberian Advances in Mathematics

, Volume 23, Issue 3, pp 210–218 | Cite as

Double exponential map on symmetric spaces

  • Yu. G. Nikonorov
Article
  • 48 Downloads

Abstract

We establish an asymptotic formula for the double exponential map operator on affine symmetric spaces. This operator plays an important role in the geometric calculus of symbols of (pseudo)differential operators on manifolds with connection, whose foundations were laid by Sharafutdinov. To obtain this result, we essentially use the structural theory of symmetric spaces and techniques of the Lie group theory. One of the key moments is an application of the Campbell-Hausdorff series in Dynkin form.

Keywords

Riemannian manifold (pseudo)differential operator on a manifold homogeneous space symmetric space Lie group 

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References

  1. 1.
    M. A. Akivis and V.V. Goldberg, “Local Algebras of Differential Quasigroups,” Bull. Amer. Math. Soc. (N.S.) 43(2), 207–226 (2006) (electronic).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    E. B. Dynkin, “On the representation by means of commutators of the series log(e x e y) for noncommutative x and y,” Mat. Sbornik N.S. 25(67) 155–162 (1949).MathSciNetGoogle Scholar
  3. 3.
    V. V. Dzhepko and Yu. G. Nikonorov, “The double exponential map on spaces of constant curvature,” Mat. Tr. 10(1), 141–153 (2007) [Siberian Adv. Math. 18, 21–29 (2008)].MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. V. Gavrilov, “The double exponential map and covariant derivation,” Sibirsk. Mat. Zh. 48(1), 68–74 (2007) [Siberian Math. J. 48 (1), 56–61 (2007)].MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. V. Gavrilov, “The Leibniz formula for the covariant derivative and some of its applications,” Mat. Tr. 13(1), 63–84 (2010) [Siberian Adv. Math. 22 (2) 80–94 (2012)].MathSciNetzbMATHGoogle Scholar
  6. 6.
    S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York-London, 1962).zbMATHGoogle Scholar
  7. 7.
    M. Kikkawa, “On local loops in affine manifolds,” J. Sci. Hiroshima Univ. Ser A-I 28, 199–207 (1964).MathSciNetzbMATHGoogle Scholar
  8. 8.
    M. Kikkawa, “Geometry of homogeneous Lie loops,” Hiroshima Math. J. 5, 141–179 (1975).MathSciNetzbMATHGoogle Scholar
  9. 9.
    M. Kikkawa, “Kikkawa loops and homogeneous loops,” Comment. Math. Univ. Carolinae 45(2), 279–285 (2004).MathSciNetzbMATHGoogle Scholar
  10. 10.
    S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, II. (Interscience Publishers, New York-London-Sydney, 1963, 1969) [(Nauka, Moscow, 1981)].Google Scholar
  11. 11.
    O. Loos, Symmetric spaces. I: General theory. II: Compact spaces and classification, (W.A. Benjamin, Inc; New York-Amsterdam, 1969) [Nauka, Moscow, 1985)zbMATHGoogle Scholar
  12. 12.
    B. Mielnik and J. Plebanski, “Combinatorial approach to Baker-Campbell-Hausdorff exponents,” Ann. Inst. H. Poincaré Sect. A (N.S.) / Physique Theorique 12(3), 215–254 (1970).MathSciNetzbMATHGoogle Scholar
  13. 13.
    M. M. Postnikov, Lie Groups and Algebras. Lectures in Geometry, Semester V (Nauka, Moscow, 1982) [in Russian].Google Scholar
  14. 14.
    M.W. Reinsch, “A simple expression for the terms in the Baker-Campbell-Hausdorff series,” J.Math. Phys. 41(4), 2434–2442 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    C. Reutenauer, Free Lie Algebras (Oxford Univ. Press, Oxford, 1993).zbMATHGoogle Scholar
  16. 16.
    F. Rouvière, “Invariant analysis and contractions of symmetric spaces I, II,” Compositio Math. 73(3), 241–270 (1990); Compositio Math. 80 (2), 111–136 (1991).MathSciNetzbMATHGoogle Scholar
  17. 17.
    L. V. Sabibin, “Loop geometries,” Mat. Zametki 12(5), 605–616 (1972) [Math. Notes 12 (5), 799–805 (1972)].MathSciNetGoogle Scholar
  18. 18.
    L. L. Sabinina, “On Kikkawa spaces,” Uspekhi Mat. Nauk 58(4) (352), 155–156 (2003) [Russian Math. Surveys. 58 (4), 796–797 (2003)].MathSciNetCrossRefGoogle Scholar
  19. 19.
    V. A. Sharafutdinov, “Geometric symbol calculus for pseudodifferential operators. I; II,” Mat. Tr. 7(2), 159–206 (2004); Mat. Tr. 8 (1), 176–201 (2005) [Siberian Adv. Math. 15 (3) 81–125 (2005); Siberian Adv. Math. 15 (4), 71–95 (2005)].MathSciNetGoogle Scholar
  20. 20.
    V. A. Sharafutdinov, Application of Geometric Symbol Calculus to Computing Heat Invariants, Preprint (2008).Google Scholar

Copyright information

© Allerton Press, Inc. 2013

Authors and Affiliations

  1. 1.Southern Mathematical Institute of the Vladikavkaz Scientific Center of RASVladikavkazRussia

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