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Acta Mathematica Hungarica

, Volume 89, Issue 1–2, pp 111–133 | Cite as

On Wagner Connections and Wagner Manifolds

  • Cs. Vincze
Article

Abstract

Let (M, E) be a Finsler manifold. A triplet (¯D, ¯h, α) is said to be a Wagner connection on M if (¯D, ¯h) is a Finsler connection, α ∈ C (M) and the axioms (W1)–(W4), formulated originally by M. Hashiguchi, are satisfied. Then ¯h is called a Wagner endomorphism on M. We establish an explicit relation between the (canonical) Barthel endomorphism of (M, E) and a Wagner endomorphism ¯h. We show that the second Cartan tensors ¯C′, ¯C b belonging to ¯h are symmetric and totally symmetric, respectively. An explicit relation between the "canonical" tensors C′, C b and the "Wagnerian" ones is also derived. We can conclude that the rules of calculation with respect to a Wagner connection are formally the same as those with respect to the classical Cartan connection. We establish some basic curvature identities concerning a Wagner connection, including Bianchi identities. Finally, we present a new, intrinsic definition as well as several tensorial characterizations of Wagner manifolds.

Keywords

Bianchi Identity Explicit Relation Finsler Manifold Curvature Identity Cartan Connection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    J. Grifone, Structure presque-tangente et connexions, I, Ann. Inst. Fourier, Grenoble, 22 (1972), 287–334.Google Scholar
  2. [2]
    J. Grifone, Structure presque-tangente et connexions, II, Ann. Inst. Fourier, Grenoble, 22 (1972), 291–338.Google Scholar
  3. [3]
    M. Hashiguchi, On Wagner's generalized Berwald space, J. Korean Math. Soc., 12 (1975), 51–61.Google Scholar
  4. [4]
    M. Hashiguchi and Y. Ichijyō, On conformal transformations of Wagner spaces, Rep. Fac. Sci. Kagoshima Univ. (Math. Phys. Chem.), 10 (1977), 19–25.Google Scholar
  5. [5]
    J. Szilasi, Notable Finsler connections on a Finsler manifold, Technical Reports No. 14/1997, Inst. of Math. and Inf., Lajos Kossuth Univ., Debrecen.Google Scholar
  6. [6]
    J. Szilasi and Cs. Vincze, On conformal equivalence of Riemann-Finsler metrics, Publ. Math. Debrecen, 52 (1998), 167–185.Google Scholar
  7. [7]
    N. L. Youssef, Semi-projective changes, Tensor, N. S., 55 (1994), 131–141.Google Scholar
  8. [8]
    V. V. Wagner, On generalized Berwald spaces, C. R. (Doklady) Acad. Sci. URSS (N.S.), 39 (1943), 3–5.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest 2000

Authors and Affiliations

  • Cs. Vincze
    • 1
  1. 1.Institute of Mathematics and InformaticsUniversity of DebrecenDebrecenHungary

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