  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.
