Finsler Spinors and Twistors in Einstein Gravity and Modifications

We present a generalization of the spinor and twistor geometry for Finsler-Cartan spaces modelled on tangent Lorentz bundles, or on (pseudo) Riemanian manifolds. Nonholonomic (Finsler) twistors are defined as solutions of generalized twistor equations determined by spin connections and frames adapted to nonlinear connection structures. We show that the constructions for local twistors can be globalized using nonholonomic deformations with 'auxiliary' metric compatible connections completely determined by the metric structure and/or the Finsler fundamental function. We explain how to perform such an approach in the Einstein gravity theory formulated in Finsler like variables with conventional nonholonomic 2+2 splitting.