ON THE EXISTENCE AND EXAMPLES OF HOMOGENEOUS GEODESICS IN GENERALIZED m-KROPINA SPACE

Seema Jangir, Gauree Shanker, Jaspreet Kaur, Laurian-Ioan Piscoran

DOI Number
https://doi.org/10.22190/FUMI230524020J
First page
289
Last page
302

Abstract


In this paper, we find a necessary and sufficient condition for a non-zero vector to be a geodesic vector in homogeneous generalized m-Kropina space. Further, we prove the existence of at least one homogeneous geodesic. However, it is conjectured that the outcomes and proofs in the case of Finsler geometry are not ideal, since general Finsler metrics are non-reversible. In Finsler geometry, the trajectory of unique homogeneous geodesic should be regarded as two geodesics with initial vectors X and -X. Hence, we construct an (n + 1)-dimensional and a 4-dimensional space to find homogeneous geodesics explicitly.


Keywords

generalized m-Kropina space, Finsler geometry, homogeneous geodesic.

Full Text:

PDF

References


V. I. Arnold: Sur la ge´ome´trie diffe´rentielle des groupes de Lie de dimension infnie et ses applications a`lhydrodynamique des fluides parfaites. Ann. Inst. Fourier (Grenoble), 16 (1960), 319–361.

M. Atashafrouz, B. Najafi and L. Pis¸coran: Left invariant (α; β)-metrics on 4-dimensional Lie groups. Facta Univ. Ser. Math. Info. 35(3) (2020), 727–740.

D. Bao, S. S. Chern and Z. Shen: An Introduction to Riemann-Finsler Geometry, GTM- 200, Springer-Verlag 2000.

S. S. Chern and Z. Shen: Riemann-Finsler Geometry, Nankai Tracts in Mathematics, Vol. 6, World Scientifc Publishers, 2005.

P. Chossat, D. Lewis, J. P. Ortega and T. S. Ratiu: Bifurcation of relative equilibria in mechanical systems with symmetry. Adv. Appl. Math. 31 (2003), 10–45.

M. Crampin and T. Mestdaga: Relative equilibria of Lagrangian systems with symmetry, J. Geom. Phys. 58 (2008), 874–887.

S. Deng: Homogeneous Finsler Spaces, Springer Monographs in Mathematics, New York, 2012.

S. Deng and Z. Hou: Invariant Finsler metrics on homogeneous manifolds. J. Phys. 37 (2004) 8245–8253.

S. Deng and Z. Hou: Invariant Randers metrics on homogeneous Riemannian manifolds. J Phys. A: Math. Gen. 37 (2004), 4353–4360.

Z. Duek: Homogeneous Randers spaces admitting just two homogeneous geodesics. Archivum Mathematicum. 55(5), 281–288.

Z. Duek: The Existence of Two Homogeneous Geodesics in Finsler Geometry, Symmetry, 11 (2019),850; doi:10.3390/sym11070850.

P. Habibi: Homogeneous geodesics in Homogeneous Randers spaces - examples. Journal of Finsler Geometry and its applications. 1(1) (2020), 89–95.

M. Hosseini and H.R. Salimi Moghaddam: On the existence of homogeneous geodesic in homogeneous Kropina spaces. Bull. Iran. Math. Soc. 46 (2020), 457–469

V. V. Kajzer: Conjugate points of left invariant metrics on Lie group. Sov. Math. 34 (1990), 32–44.

K. Kaur and G. Shanker: On the geodesics of a homogeneous Finsler space with a special (α; β)-metric. J. Fins. Geo. Appl. 1(1) (2020), 26–36.

O. Kowalski: Generalized Symmetric spaces Lecture Notes in Math. Vol. 805, Springer Verlag, Berlin-Heidelberg-New York, 1980.

O. Kowalski and J. Szenthe: On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata, 81 (2000), 209–214.

O. Kowalski and Z. Vla´sˇek: Homogeneous Riemannian manifolds with only one homogeneous geodesic. Publ. Math. Debr. 62 (3-4) (2003), 437–446.

V. K. Kropina: On projective two-dimensional Finsler spaces with a special metric, Trudy Sem. Vektor. Tenzor. Anal. 11 (1961) 277–292, (in Russian).

E. A. Lacomba: Mechanical Systems with Symmetry on Homogeneous Spaces. Trans. Amer. Math. Soc. 185 (1973), 477–491.

D. Latiffi: Homogeneous geodesics in homogeneous Finsler spaces, J. Geom. Phys. 57 (2007), 1421–1433.

J. P. Ortega and T. S. Ratiu: Stability of Hamiltonian relative equilibria. Nonlinearity. 12 (1999), 693–720.

G. W. Patrick: Relative Equilibria of Hamiltonian Systems with Symmetry: Linearization, Smoothness, and Drift. J. Nonlinear Sci. 5 (1995), 373–418.

M. Parhizkar and H. R. Salimi Moghaddam: Geodesic Vector Fields of Invariant (α; β)- Metrics on Homogeneous Spaces. Inter. Elec. Jour. Geo. 6 (2) (2013), 39–44.

S. Rani and G. Shanker: On S-curvature of homogeneous Finsler spaces with Randers changed square metric. Facta Univer. Ser. Mathe. Info. 35 (3) (2020), 673–691.

G. Z. To´th: On Lagrangian and Hamiltonian systems with homogeneous trajectories. J. Phys. A: Math. Theor. 43 (2010), 385206 (19pp).

Z. Yan and S. Deng: Existence of homogeneous geodesics on homogeneous Randers spaces. Hou. J. Math. 44 (2) (2018), 481–493.




DOI: https://doi.org/10.22190/FUMI230524020J

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)