  While there have been extensive studies regarding the theory of composition
operators in standard Bergman spaces, there have not been many results
pertaining to large Bergman spaces due to a lack of useful tools. In this
paper, we give the characterizations of the compact differences of composition
operators in Bergman spaces with the exponential type weight using a newly
defined Riemannian distance. Furthermore, we give a sufficient condition for
the question when two composition operators lie in the same component.
