Morse指标和闭测地线的稳定性

Morse index and the stability of closed geodesics

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.