The Kastler-Kalau-Walze type theorem for 6-dimensional manifolds with boundary

In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume ${\rm Vol}_{6}^{(1,3)}$ for 6-dimensional spin manifolds with boundary and the gravity on boundary is derived by the noncommutative residue associated with Dirac operators.For 6-dimensional manifolds with boundary, we also get a Kastler-Kalau-Walze type theorem for a general fourth order operator.