Morse指标和闭测地线的稳定性  

Morse index and the stability of closed geodesics

作　　者：胡锡俊[1] 孙善忠[2] 

机构地区：[1]山东大学数学学院,济南250100 [2]首都师范大学数学科学学院,北京100048

出　　处：《中国科学：数学》2010年第5期415-420,共6页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:10801127;10731080)资助项目

摘　　要：令ind(c)为n+1维Riemann流形M上闭测地线c的Morse指标.我们证明:对于闭测地线c,如果它是定向的且n+ind(c)是奇数,或它是非定向的且n+ind(c)为偶数,则c不稳定.Poincaré的一个著名定理说Riemann面上的极小闭测地线是不稳定的,我们的结果是该结论的一个推广.Let ind(c)be the Morse index of a closed geodesic c in an(n+1)-dimensional Riemannian manifold M.We prove that an oriented closed geodesic c is unstable if n+ind(c)is odd and a non-oriented closed geodesic c is unstable if n+ind(c)is even.Our result is a generalization of the famous theorem due to Poincaré which states that the closed minimizing geodesic on a Riemann surface is unstable.

关 键 词：闭测地线 稳定性 MORSE指标 

分 类 号：O186.12[理学—数学]