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This text is the extended version of a talk given at 6th Meeting ofIntegrable Systems and Quantum Filed Theory at Peyresq hold from June 10 2006to June 17, 2006 at Peyresq, France. The goal of this lecture is to give abrief introduction to Cartan-K\"ahler's theory. As examples to the applicationof this theory, we choose the local isometric and conformal embedding. Weprovide lots of details and explanations of the calculation and the tools used.

Cartan-K\"ahler Theory and Applications to Local Isometric and Conformal Embedding