The geometry of Ingarden spaces

The Randers spaces RFn were introduced by R. S. Ingarden [8]. They are considered as Finsler spaces F-n = (M, alpha + beta) equipped with the Cartan nonlinear connection. In the present paper we define and study what we call the Ingarden spaces, IFn, as Finsler spaces F-n = (M, alpha + beta) equipped with the Lorentz nonlinear connection. The spaces RP and IFn are completely different. For IFn we discuss: the variational problem, Lorentz nonlinear connection, canonical N-metrical connection and its structure equations, the Cartan 1-form omega, the electromagnetic 2-form F and the almost symplectic 2-form theta. The formula domega = -F + theta is established. It has as a consequence the generalized Maxwell equations. Finally, the almost Hermitian model of IFn is constructed.