Geodesics are the curves that represent the shortest path between two points on a surface or in a given space, often described in the context of Riemannian and sub-Riemannian geometries. They generalize the concept of straight lines to curved spaces, providing critical insights into the geometry and topology of various manifolds. Understanding geodesics helps in analyzing distances, minimizing paths, and exploring the intrinsic geometric structure of spaces, particularly in more complex settings like sub-Riemannian manifolds and variational problems involving mean curvature.
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