Abstract
The notions of real Hamilton and Cartan spaces were defined and exhaustively studied by R.Miron ([12]) who established the ℒ—duality between Lagrange and Hamilton spaces, particularly between Finsler and Cartan spaces.
The aim of this note is to investigate the measure in which the geometry of complex Hamilton space is recovered by Legendre transformation from one of complex Lagrange space. In this sense, it is of interest to obtain the image by Legendre transformation of main geometrical objects from a complex Lagrange space on a complex Hamilton spaces, called ℒ—dual of the first.
In the paper it is proved that there exists only one pair of complex nonlinear connections which correspond by ℒ—duality. A deep analysis of the ℒ—dual of N—complex linear connections, and in particular of the metric connections is done.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abate, M. and Patrizio, G., Finsler metrics — A global approach, Lecture Notes in Math., 1591, Springer-Verlag, 1994.
Abraham, R., Marsden, J.E. and Ratiu, T., Manifolds, Tensor Analysis and Apppl., Springer-Verlag, second edition, 1988.
Aikou, T., On complex Finsler manifolds, Rep. of Kagoshima Univ. No. 24, (1991), 9–25,
Arnold, V.I., Mathematical Methods of Classical Mechanics, SpringerVerlag, 1978.
Kobayashi, S., Negative vector bundles and complex Finsler structures, Nagoya Math. J., 57 (1975), p.153–166.
Faran, J., The equivalence problem for complex Finsler Hamiltonian, Contemporary Math., (1996), 133–144.
Hrimiuc, D. and Shimada, H., On the ℒ-duality between Lagrange and Hamilton geometry, Nonlinear Word, 3 (1996), 613–641.
Miron, R. and Anastasiei, M., Vector bundles, Lagrange spaces, Application to the Theory of Relativity, Balkan Press, Bucuresti, 1997.
Miron, R. and Anastasiei, M. The Geometry of Lagrange Spaces; Theory and Applications, Kluwer Acad.Publ., no. 59, FTPH, 1994.
Miron, R., Hamilton geometry, Analele St. Univ. Iasi, s.I,a Mat., 35 (1989), 38–85.
Miron, R., Ianus, S. and Anastasiei, M., The geometry of the dual of a vector bundle, Publ.de L’Institut Mathematique, Nouvelle serie, t. 46 (60) (1989), 145–162.
Miron, R., Hrimiuc, D., Shimada, H., and Sabau, S., The Geometry of Hamilton and Lagrange Spaces, Kluwer Acad. Publ., no.118, FTPH, 2001.
Munteanu, G., Complex Lagrange spaces, Balkan J. of Gem. and its Appl., Vol 3, no 1 (1998), 61–71.
Munteanu, G., Complex Hamilton spaces, Algebras, Groups and Geometries, 17 (2000), 293–302.
Munteanu, G., On Chern-Lagrange complex connection, Steps in Diff. Geom. Conf. Debrecen, (2000), 237–242.
Munteanu, G., The geometry of complex Hamilton spaces, (to appear).
Oproiu, V. and Papaghiuc, N., On differential geometry of the Legendre transformation, Rend. Sem. Sc. Univ. Cagliari, 57, nr. 1 (1987), 35–49.
Vaisman, I., Symplectic Geometry and Secondary Characteristic Classes, Birkhäuser, Boston-Basel, 1987.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Munteanu, G. (2003). ℒ-Dual Complex Lagrange and Hamilton Spaces. In: Anastasiei, M., Antonelli, P.L. (eds) Finsler and Lagrange Geometries. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0405-2_15
Download citation
DOI: https://doi.org/10.1007/978-94-017-0405-2_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6325-0
Online ISBN: 978-94-017-0405-2
eBook Packages: Springer Book Archive