Generalized Lagrange Transforms: Finsler Geometry Methods and Deformation Quantization of Gravity

We propose a natural Fedosov type quantization of generalized Lagrange models and gravity theories with metrics lifted on tangent bundle, or extended to higher dimension, following some stated geometric/ physical conditions (for instance, nonholonomic and/or conformal transforms to some physically important metrics or mapping into a gauge model). Such generalized Lagrange transforms define canonical nonlinear connection, metric and linear connection structures and model almost Kähler geometries with induced canonical sympletic structure and compatible affine connection. The constructions are possible due to a synthesis of the nonlinear connection formalism developed in Finsler and Lagrange geometries and deformation quantization methods.