Lagrange Geometry via Complex Lagrange Geometry

Asking that the metric of a complex Finsler space should be strong convex, Abate and Patrizio ([1]) associate to the real tangent bundle a real Finsler metric for which they analyze the relation between Cartan (real) connection of the obtained space and the real image of Chern-Finsler complex connection. Following the same ideas, in the present paper we shall deal with the more general case of a complex Lagrange space (M, L). As distinct from these authors, we shall associate to the Hermitian metric gij̄(z, η) of a complex Lagrangian L its real representation R gab (x, y). The obtained real space (M, R gab) is a generalized Lagrange space ([10]). Furthermore, the possibility of its reduction to one real Lagrange space, in particular the Finsler one, is studied. A comparative analysis of the elements of Lagrange geometry ([10]): nonlinear connection, N−linear connection, metric canonical connection, and so on, and their corresponding real image from the complex Lagrange geometry ([11]) is made. AMS Mathematics Subject Classification (2000): 53B40, 53C60