We will consider the Cauchy problem for nonlinear hyperbolic equations of second order with smooth data. It is well known that the Cauchy problem has a smooth solution in a neighbourhood of the initial curve. But it might fail to admit a smooth solution in the whole space. This means that singularities appear generally in finite time. We are interested in the global theory. Therefore our problem is how to extend the solution after the appearance of singularities. For this purpose, we will first lift the solution surface into cotangent space so that the singularities would disappear, and we will construct globally a geometric solution there. Next we will project it to the base space. In this procedure we will meet the singularities of smooth mappings.