The Geometry of Geodesics in Riemannian Manifolds

Geodesics are the shortest or most natural paths between points on a curved surface or space, similar to straight lines in flat geometry. In Riemannian manifolds, which are spaces with curvature, geodesics are curves that locally minimize distance, determined by the manifold’s shape. These paths appear as "straight" within the context of the space's geometry, taking into account curvature. They provide a way to understand how objects move or connect points in complex, curved environments, extending the familiar idea of straight lines into more intricate, curved settings.