The Kastler-Kalau-Walze Type Theorem For Witten Deformation

In this thesis,we compute lower dimensional volumes Vol₄^1,1and Vol₆^2,2for 4,6-dimensional spin manifolds with boundary respectively,and get assosi-ated Kastler-Kalau-Walze type theorems.We also give theoritic explaination of the gravitational action for 4,6 dimensional manifolds with boundary by these noncommutative residues.