Noncommutative residue and sub-Dirac operators for foliations

Abstract

In this paper, we define lower dimensional volumes associated with 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] ×f M3.