摘要
设En是在0∈R~n的C函数芽环,M是E_n中唯一的极大理想.如果f∈M~2且其二阶Hessain是非退化的,则f同构于它的二阶Hessain,这就是著名的Morse引理,本文将讨论两个变元的C~∞函数芽,得到:(1)若f∈M~3∈E_(xy),且其三阶Hessain是非退化的,则f同构于它的三阶Hessain。 (2)若f∈M_4∈E_(xy),其四阶Hessain是非退化的,则f同构于它的四阶Hessain.显然,这是Morse引理的一种推广.
Assume that E_n is the ring of the C~∞ function germs at 0∈ R^n,M is the unique maximal ideal in E_n. If f ∈M^2 and its quadratic Hessain is non-degenerate, then f is isomorphic to its quadratic Hessain. This is famous Morse lemma. In this paper, We will discuss C funtion germs in two variables. The results show that (1) If f∈M^3(E_(xy) and its cubic Hessain is non-degenerate, then f is isomorphic to its cubic Hessain. (2) If f∈M^4( M E_(xy) and its Hessain of degree 4 is non-degenerate,then f is isomorphic to its Hessain of degree 4. Obviously, this is a generalization of Morse lemma.