The length of curves is a measure of the distance along a curve, calculated by integrating the speed of a parameterized curve over its domain. This concept is crucial when discussing induced metrics on submanifolds and the Riemannian distance function, as it provides a way to understand how lengths are defined and computed in various geometric contexts. By understanding the length of curves, one can grasp how distances are represented in curved spaces, allowing for deeper insights into the geometry of these manifolds.
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