Gauss-Bonnet-Chern formula is a remarkably fundamental result which builds a connection between differential geometry and topology. It has been successfully generated on Poincaré-Einstein manifolds through renormalized curvature integral by Albin. With the aid of ambient space construction, we give a general formulism for renormalized curvature integral. In particular, we give a reformulation of Gauss-Bonnet-Chern formula on Poincaré-Einstein manifolds, which provides some interesting applications. This work is a joint work with Jeffrey S. Case, Ayush Khaitan, Yueh-Ju Lin and Aaron J. Tyrrell.
