Differential Forms and the Noncommutative Residue for Manifolds with   Boundary in the Non-product Case
  In this paper, for an even dimensional compact manifold with boundary which has the non-product metric near the boundary, we use the noncommutative residue to define a conformal invariant pair. For a 4-dimensional manifold, we compute this conformal invariant pair under some conditions and point out the way of computations in the general. 
Yong Wang