In this paper, let m ≥ 1 be an integer, M be an m-dimensional compact Riemannian manifold. Firstly the linearized Poincare map of the Lagrangian system at critical point x $${d \over {dt}}{L_q}\left( {t,x,\dot x} \right) - {L_p}\left( {t,x,\dot x} \right) = 0$$ is explicitly given, then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index, finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.

中文翻译：

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黎曼流形上拉格朗日系统的指数和稳定性

本文中，设m≥1为整数，M为m维紧黎曼流形。首先是拉格朗日系统在临界点 x $${d \over {dt}}{L_q}\left( {t,x,\dot x} \right) - {L_p}\left( {t ,x,\dot x} \right) = 0$$ 是明确给出的，那么我们证明无论流形 M 是否可定向，x 的 Morse 指数和 Maslov 型指数是明确定义的，这使得没有诉诸幺正平凡化，建立Morse指数与Maslov型指数的关系，最终推导出M的临界点和方向不稳定性判据，得到两个Maslov型指数的公式。