DOI QR코드

DOI QR Code

LICHNEROWICZ CONNECTIONS IN ALMOST COMPLEX FINSLER MANIFOLDS

  • Published : 2005.05.01

Abstract

We consider the connections $\nabla$ on the Rizza manifold (M, J, L) satisfying ${\nabla}G=0\;and\;{\nabla}J=0$. Among them, we derive a Lichnerowicz connection from the Cart an connection and characterize it in terms of torsion. Generalizing Kahler condition in Hermitian geometry, we define a Kahler condition for Rizza manifolds. For such manifolds, we show that the Cartan connection and the Lichnerowicz connection coincide and that the almost complex structure J is integrable.

Keywords

Finsler metric;almost complex manifold;Rizza manifold;Cartan connection;Kahler condition

References

  1. Y. Ichijyo, Almost Hermitian Finsler manifolds and nonlinear connections, Conf. Semin. Mat. Univ. Bari 215 (1986), 1-13
  2. D. Bao, S. S. Chern, and Z. Shen, An Introduction to Riemannian-Finsler geom- etry, Grad. Texts in Math. 2000
  3. M. Abate and G. Patrizio, Finsler metric-A global approach, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1591 (1994)
  4. E. Heil, A relation between Finslerian and Hermitian metrics, Tensor (N.S.) 16 (1965), 1-3
  5. Y. Ichijyo, Finsler metrics on almost complex manifolds, Geometry Conference (Parma, 1988), Riv. Mat. Univ. Parma (4) 14 (1988), 1-28
  6. G. B. Rizza, Strutture di Finsler sulle varieta quasi complesse, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei. (8) 33 (1962), 271-275
  7. G. B. Rizza, Strutture di Finsler di tipo quasi Hermitiano, Riv. Mat. Univ. Parma (2) 4 (1963), 83-106
  8. S. Kobayashi, Almost complex manifolds and hyperbolicity, Results Math. 40 (2001), 246-256 https://doi.org/10.1007/BF03322709
  9. S. Kobayashi, Natural connections in almost complex manifolds, Contemp. Math. 332 (2003), 175-178
  10. N. Lee and D. Y. Won, Connections in almost complex Finsler manifolds and hyperbolicity, submitted, 2004
  11. M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Japan, 1986

Cited by

  1. An almost paracontact structure on a Rizza manifold vol.349, pp.11-12, 2011, https://doi.org/10.1016/j.crma.2011.06.005