Nonlinear Connections on Gerbes, Clifford-Finsler Modules, and the Index Theorems

The geometry of nonholonomic bundle gerbes, provided with nonlinear connection structure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah-Singer index formulas. There are considered certain applications in modern gravity and geometric mechanics of such Clifford-Lagrange/ Finsler gerbes and their realizations as nonholonomic Clifford and Riemann-Cartan modules.