The Cauchy problem has been considered for nonlinear hyperbolic equations of second order with smooth data. It is well known that the Cauchy problem has a smooth solution in a neigh bourhood of the initial curve. But it might fail to admit a smooth solution in the whole space. This means that singularities apear generally in finite time. The author was interested in the global theory. Therefore, the problem is how to extend the solution after the appearance of singularities. for this purpose, the author has lift the solution surface into contangent space to that the singularities would disappear, and construct globally a geometric solution there. After that the author projected it to the base space. The final results are showing the singularities of smooth mappings.