Lower dimensional volumes and the Kastler-Kalau-Walze type theorem for Manifolds with Boundary

In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume ${\mathrm{V}\mathrm{o}\mathrm{l}}^{(2,2)}$ for 5-dimensional and 6-dimensional spin manifolds with boundary and we also get the Kastler-Kalau-Walze type theorem in this case.