Riemann-Finsler and Lagrange Gerbes and the Aiyah--Singer Theorems

Abstract

In this paper, nonholonomic gerbes will be naturally derived for manifolds and vector bundle spaces provided with nonintegrable distributions (in brief, nonholonomic spaces). An important example of such gerbes is related to distributions defining nonlinear connection (N-connection) structures. They geometrically unify and develop the concepts of Riemann-Cartan manifolds and Lagrange-Finsler spaces. The obstruction to the existence of a spin structure on nonholonomic spaces is just the second Stiefel-Whitney class, defined by the cocycle associated to a $\mathbb{Z} /2$ gerbe, which is called the nonholonomic spin gerbe. The nonholonomic gerbes are canonically endowed with N-connection, Sasaki type metric, canonical linear connection connection and (for odd dimension spaces) almost complex structures. The study of nonholonomic spin structures and gerbes have both geometric and physical applications. Our aim is to prove the Atiyah--Singer theorems for such nonholonomic spaces.