This paper presents global solutions of adiabatic accretion flows with isothermal shocks in Kerr black hole geometry. It is known that in the previously studied cases, where the flow including the shock is either entirely adiabatic or entirely isothermal, there can be no more than one stable shock solution, and the solution can only be of a-x type. However, the solution topology in the present case shows remarkable new characteristics: for the same flow parameters there can be two stable shock solutions satisfying physical boundary conditions, and the solution can be of three types, namely a-x, x-a and a-a type. In addition, shocks in the present case occur for a parameter region different from that for Rankine-Hugoniot shocks. These results greatly increase the possibilities of shock formation in astrophysical flows. It is also significant that the effects of frame-dragging of a rapid Kerr black hole on the shock formation are discovered. Finally, a brief comparison is made between shocked inviscid flows and two types of shock-free viscous flows, namely those of Shakura & Sunyaev and Narayan & Yi, and some comments are made about the fact that numerous authors who have studied transonic global solutions of accretion flows have found no shocks.