Distinguished Connections on Finsler Algebroids

Abstract

Considering the prolongation of a Lie algebroid, the authors introduce Finsler algebroids and present important geometric objects on these spaces. Important endomorphisms like conservative and Barthel, Cartan tensor and some distinguished connections like Berwald, Cartan, Chern-Rund and Hashiguchi are introduced and studied.

Keywords

Chern-Rund connection / Distinguished connections / Finsler algebroid / Hashiguchi connection / Lie algebroid

References

[1] Bao D, Chern S, Shen Z. An Introduction to Riemann-Finsler Geometry, 2000, New York: Springer-Verlag

[2] Cortez J, de Leon M, Marrero J C A survey of Lagrangian mechanics and control on Lie algebroids and groupoids. Int. J. Geom. Methods Mod. Phys., 2006, 3(3): 509-558

[3] Grabowski J, Urbański P. Tangent and cotangent lift and graded Lie algebra associated with Lie algebroids. Ann. Global Anal. Geom., 1997, 15: 447-486

[4] Grabowski J, Urbański P. Lie algebroids and Poisson-Nijenhuis structures. Rep. Math. Phys., 1997, 40: 195-208

[5] de León M, Marrero J C, Martínez E. Lagrangian submanifolds and dynamics on Lie algebroids. J. Phys. A: Math. Gen., 2005, 38: 241-308

[6] Martinez E. Lagrangian mechanics on Lie algebroids. Acta Appl. Math., 2001, 67: 295-320

[7] Mestdag, T., A Lie algebroid approach to Lagrangian systems with symmetry, Diff. Geom. Appl., 2005, 523–535.

[8] Peyghan, E., Berwald-type and Yano-type connections on Lie algebroids, Int. J. Geom. Methods Mod. Phys., 12 (10), 2015, 36 pages.

[9] Popescu L. The geometry of Lie algebroids and applications to optimal control. Annals. Univ. Al. I. Cuza, Iasi, Series I, Math., 2005, LI: 155-170

[10] Popescu L. Aspects of Lie algebroids geometry and Hamiltonian formalism. Annals Univ. Al. I. Cuza, Iasi, Series I, Math., 2007, LIII(supl.): 297-308

[11] Popescu L. A note on Poisson-Lie algebroids. J. Geom. Symmetry Phys., 2008, 12: 63-73

[12] Pradines J. Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux. C. R. Acad. Sci. Paris, 1967, 264(A): 245-248

[13] Szakál S, Szilasi J. A new approach to generalized Berwald manifolds I. SUT J. Math., 2001, 37: 19-41

[14] Szakál S, Szilasi J. A new approach to generalized Berwald manifolds I. Publ. Math. Debrecen, 2002, 60: 429-453

[15] Szilasi J. A setting for spray and finsler geometry, Handbook of Finsler Geometry, 2003, Dordrecht: Kluwer Academic Publishers 1183-1426

[16] Szilasi J, Győry A. A generalization of Weyl’s theorem on projectively related affine connections. Rep. Math. Phys., 2004, 54: 261-273

[17] Szilasi J, Tóth A. Conformal vector fields on Finsler manifolds. Comm. Math., 2011, 19: 149-168

[18] Vacaru S I. Nonholonomic algebroids, Finsler geometry, and Lagrange-Hamilton spaces. Mathematical Sciences, 2012, 6: 1-33

[19] Vacaru, S. I., Almost Kähler Ricci flows and Einstein and Lagrange-Finsler structures on Lie algebroids, 2013, arXiv: submit/0724182 [math-ph].

[20] Weinstein A. Lagrangian mechanics and grupoids. Fields Institute Communications, 1996, 7: 207-231