Classification Of Germs Of C~∞ Real Functions With Corank 2 And Codimension 6

Classification of germs of C∞real functions is an important issue of the singularity theory. The classification of C∞real function germs whose codimension less than or equal to five had been done by R.Thom. In [6], classification of C∞function germs for corankf≠2 and codimf=7 is an beneficial and interesting exploration and attempt for C∞function germs in higher codimension. The research of [6] shows that classification of C∞function germs for corankf≠2 and codimf=7 is resulted in classification of germs in one variable with codimension 7. In essence, this is a result of Thom's classification.The aim of this thesis is to study the classification problem of C∞real function germs with corank 2 and codimension 6.the classification of C∞real function germs with corank 2 and codimension 6 in En will be converted into the classification of germs of codimension 6 in two variables by using splitting lemma. And then, we propose and prove the normal forms for C∞real function germs with corank 2 and codimension 6 by means of Nakayama lemma and finite determined theorem as well as the relational conclusions,which is an important complementarity and advance of R.Thom's classification. The results are as follows.Theorem A: Let g ( x , y )∈m3(2) be of codimension 6 in E2 , then g ( x , y ) is isomorphic to one (and only one )of the following germs: (ⅰ) x 3 + y4 (ⅱ) x 3 - y4 (ⅲ) x 2 y + y5 (ⅳ) x 2 y - y5So we get the classification of C∞real function germs with corank 2 and codimension 6 as follows:Theorem B: Let ( )f x∈En be of codimension 6 and corank 2 in En , then f ( x ) is isomorphic to one (and only one )of the follows germs: Whereεi=±1,"±"depend on the sign of coefficient of xn .