When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves’ lengths do we really need to know? It is a result of Duchin, Leininger, and Rafi that any flat metric induced by a unit-norm quadratic differential is determined by its marked simple length spectrum. We generalize the notion of simple curves to that of $q$-simple curves, for any positive integer $q$, and show that the lengths of $q$-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a $q$-differential.

中文翻译：

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来自$ q $-差分的平面度量标准的长度谱

当曲面的几何结构由曲线的长度决定时，自然会问：我们真正需要知道哪些曲线的长度？是Duchin，Leininger和Rafi的结果，由单位范数二次微分引起的任何平坦度量均由其明显的简单长度谱确定。对于任何正整数$ q $，我们将简单曲线的概念推广到$ q $-简单曲线的概念，并表明$ q $-简单曲线的长度足以确定由引起的非正弯曲欧几里德锥度$ q $的差异。