We extend some results of nonsmooth analysis from the Euclidean context to the Riemannian setting. Particularly, we discuss the concepts and some properties, such as the Clarke generalized covariant derivative, upper semicontinuity, and Rademacher theorem, of locally Lipschitz continuous vector fields on Riemannian settings. In addition, we present a version of the Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields. For mild conditions, we establish the well-definedness and local convergence of the sequence generated using the method in a neighborhood of a singularity.

中文翻译：

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寻找黎曼流形上一类特殊局部 Lipschitz 连续向量场的奇点的牛顿法

我们将非光滑分析的一些结果从欧几里得上下文扩展到黎曼设置。特别地，我们讨论了黎曼设置下局部 Lipschitz 连续向量场的概念和一些性质，例如克拉克广义协变导数、上半连续性和拉德马赫定理。此外，我们提出了牛顿方法的一个版本，用于寻找一类特殊的局部 Lipschitz 连续向量场的奇点。对于温和的条件，我们在奇异点的邻域中建立使用该方法生成的序列的明确定义和局部收敛。