Noncommutative Residue and sub-Dirac Operators for Foliations

Abstract: In this paper, we define lower dimensional volumes associated to sub-Dirac operators for foliations. In some cases, we compute these lower dimensional volumes. We also prove the Kastler-Kalau-Walze type theorems for foliations with or without boundary. As a corollary, we give an explanation of the gravitational action for the Robertson-Walker space $[a, b]\times_{f}M^{3}$.