Cartan-Kähler Theory and Applications to Local Isometric and Conformal Embedding

This text is the extended version of a talk given at 6th Meeting of Integrable Systems and Quantum Filed Theory at Peyresq held from June 10 2006 to June 17, 2006 at Peyresq, France. The goal of this lecture is to give a brief introduction to Cartan-Kähler\u27s theory. As examples to the application of this theory, we choose the local isometric and conformal embedding. We provide lots of details and explanations of the calculation and the tools used