Nullity Distributions Associated with Hashiguchi Connection

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-184
Published: Aug 1, 2025
Cite as:
A. Soleiman, S. G. Elgendi, Nullity Distributions Associated with Hashiguchi Connection, Int. J. Anal. Appl., 23 (2025), 184.

Main Article Content

A. Soleiman, S. G. Elgendi

Abstract

In this paper, we employ the Klein-Grifone formalism to study the nullity distributions associated with the curvature tensors of the Hashiguchi connection. We establish several important results regarding the nullity distribution \(\mathcal{N}_{R^\star}\) of the h-curvature tensor \(\overset{*}{R}\), demonstrating that \(\mathcal{N}_{R^\star}\) is completely integrable and that its corresponding foliation consists of auto-parallel leaves. An illustrative example shows that the nullity distribution \(\mathcal{N}_{P^\star}\), associated with the hv-curvature tensor \(\overset{*}{P}\), is not generally completely integrable. Furthermore, we determine necessary and sufficient conditions under which \(\mathcal{N}_{P^\star}\) becomes completely integrable.

Article Details

References

  1. H. Akbar-Zadeh, Espaces de Nullité de Cértains Opérateurs en Géométrie des Sous-Variétés, C. R. Acad. Sci. Paris, Sér. A 274 (1972), 490–493.
  2. H. Akbar-Zadeh, Espaces de Nullité en Géométrie Finslérienne, Tensor N. S. 26 (1972), 89–101.
  3. B. Bidabad, M. Rafie-Rad, On K-Nullity Foliations in Finsler Geometry and Completeness, arXiv:1101.1496 (2011). http://arxiv.org/abs/1101.1496v1.
  4. E. Cartan, Leçons sur la Géométrie des Espaces de Riemann, Gauthier-Villars, Paris, 1928.
  5. S. Chern, N.H. Kuiper, Some Theorems on the Isometric Imbedding of Compact Riemann Manifolds in Euclidean Space, Ann. Math. 56 (1952), 422–430. https://doi.org/10.2307/1969650.
  6. A. Frà ˝ulicher, A. Nijenhuis, Theory of Vector-Valued Differential Forms, Indag. Math. (Proc.) 59 (1956), 338–350. https://doi.org/10.1016/s1385-7258(56)50046-7.
  7. J. Grifone, Structure Presque-Tangente Et Connexions, I, Ann. Inst. Fourier 22 (1972), 287–334. http://www.numdam.org/item?id=AIF_1972__22_1_287_0.
  8. J. Grifone, Structure presque-tangente et connexions, II, Ann. Inst. Fourier 22 (1972), 291–338. https://www.numdam.org/item/AIF_1972__22_3_291_0.
  9. J. Klein, A. Voutier, Formes Extérieures Génératrices de Sprays, Ann. Inst. Fourier 18 (1968), 241–260. https://www.numdam.org/item/AIF_1968__18_1_241_0.
  10. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II, Interscience, New York, 1969.
  11. R. Maltz, The Nullity Spaces of the Curvature Operator, Cah. Topol. Géom. Différ. Catég. 8 (1965), 1–20. https://www.numdam.org/item/?id=CTGDC_1966__8__A4_0.
  12. J. Szilasi, C. Vincze, A New Look at Finsler Connections and Special Finslermanifolds, Acta Math. Acad. Paedagog. Nyhazi. 16 (2000), 33–63.
  13. N.L. Youssef, Distribution de Nullité du Tensor de Courbure d’une Connexion, C. R. Acad. Sci. Paris, Sér. A 290 (1980), 653–656.
  14. N.L. Youssef, Sur les Tenseurs de Courbure de la Connexion de Berwald et ses Distributions de Nullité, Tensor, N. S. 36 (1982), 275–280. https://cir.nii.ac.jp/crid/1572543024816947840.
  15. S.G. Elgendi, A. Soleiman, The Existence and Uniqueness of Hashiguchi Connection in KG-Approach, arXiv:2506.09739 (2025). http://arxiv.org/abs/2506.09739v1.
  16. N.L. Youssef, S. Elgendi, New Finsler Package, Comput. Phys. Commun. 185 (2014), 986–997. https://doi.org/10.1016/j.cpc.2013.10.024.
  17. N.L. Youssef, S.G. Elgendi, Nullity Distributions Associated to Chern Connection, Publ. Math. Debrecen 88 (2016), 235–248.
  18. N.L. Youssef, S.G. Elgendi, Existence and Uniqueness of Chern Connection in the Klein-Grifone Approach, J. Dyn. Syst. Geom. Theor. 18 (2020), 193–209. https://doi.org/10.1080/1726037x.2020.1856337.
  19. N.L. Youssef, A. Soleiman, S.G. Elgendi, Nullity Distributions Associated to Cartan Connection, Indian J. Pure Appl. Math. 45 (2014), 213–238. https://doi.org/10.1007/s13226-014-0060-0.