Abstract: In this paper, we discuss the invariant complete metric on the Cartan-Hartogs domain of the fourth type. Firstly,we find a new invariant complete metric, and prove the equivalence between Bergman metric and the new metric; Secondly, the Ricci curvature of the new metric has the super bound and lower bound; Thirdly,we prove that the holomorphic sectional curvature of the new metric has the negative supper bound; Finally, we obtain the equivalence between Bergman metric and Einstein-Kahler metric on the Cartan-Hartogs domain of the fourth type.

Key words: Cartan-Hartogs domain, equivalence of invariant metric, Bergman metric;
Einstein-K¨ahler metric