Complex Lie algebroids and Finsler manifold in time-dependent fractal dimension and their associated decomplexifications

Abstract

We study the variational integration problem for Lie algebroids and Finsler manifold in time-dependent fractal dimension. The theory in general is complexified and complex geodesics are obtained accordingly. However, decomplexification or the transition from C→R is possible if the action integral by itself is complexified. Their implications in Finsler manifold were also discussed and several consequences as energy, length and Hamilton's equations are explored and discussed in details.