Morse引理的一个推广

A Generalization of Morse Lemma

中文关键词:  C^∞函数芽  同构  三阶Hessain  四阶Hessain.

英文关键词:C^∞ function germs  isomorphic  cubic Hessain  Hessain of degree 4.

中文摘要:

      设E_n是在0∈Rⁿ的C^∞函数芽环,M是En中唯一的极大理想.如果f∈M²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∈M³?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∈M⁴?E_xy

英文摘要:

      Assume that E_n is the ring of the C^∞ function germs at 0∈ Rⁿ,M is the unique maximal ideal in E_n. If f ∈M² 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∈M³?E_xy and its cubic Hessain is non-degenerate, then f is isomorphic to its cubic Hessain. (2) If f∈M⁴?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.