The Geometry of Lagrange Spaces: Theory and Applications

Anholonomic frames, generalized Killing equations, and anisotropic Taub-NUT spinning spaces

“…[3,[15][16][17]20]). A N -connection induces a global decomposition of the 5D pseudo-Riemannian spacetime into holonomic (horizontal, h) and anholonomic (vertical, v) directions.…”

“…In a preliminary form the concept of N -connections was applied by E. Cartan in his approach to Finsler geometry [18] and a rigorous definition was given by Barthel [19] (Ref. [20] gives a modern approach to the geometry of N -connections, and to generalized Lagrange and Finsler geometry, see also Ref. [16] for applications of N -connection formalism in supergravity and superstring theory).…”

“…(the d-torsion is computed by substituting the h-v-components of the canonical dconnection (19) and anholonomic coefficients (18) into the formula for the torsion coefficients (20) …”

Jet geometrical objects depending on a relativistic time

“…For such a reason, a lot of authors (Asanov [2], Saunders [11], Vondra [12] and many others) studied the differential geometry of the 1-jet spaces. Going on with the geometrical studies of Asanov [2] and using as a pattern the Lagrangian geometrical ideas developed by Miron, Anastasiei or Bucȃtaru in the monographs [6] and [3], the author of this paper has developed the Riemann-Lagrange geometry of 1-jet spaces [7] which is very suitable for the geometrical study of the relativistic non-autonomous (rheonomic) Lagrangians, that is of Lagrangians depending on an usual "relativistic time" [8] or on a "relativistic multi-time" [7], [9].…”

“…In what follows we try to expose the main geometrical and physical aspects which differentiate the both geometrical theories: the jet relativistic nonautonomous Lagrangian geometry [8] and the classical non-autonomous Lagrangian geometry [6].…”