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Dragonfall
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What sort of structure must a manifold possesses in order to talk about minimal geodesics between two points on it? When can we extend the minimal geodesics indefinitely?
Dragonfall said:Given a minimal geodesic between two points, is there a unique way to extend it past the points? Indefinitely?
Dragonfall said:Can a geodesic really not fill a manifold for dimension reasons? What about space-filling curves?
Dragonfall said:Can a geodesic really not fill a manifold for dimension reasons? What about space-filling curves? Like Peano's space filling curve. A geodesic can self-intersect, and having it fill a space doesn't make it a homeomorphism.
The billiard ball problem is actually what motivated me to ask this. But I am still too uncomfortable with manifolds to fully understand your answers. I'm going to work on learning the basics some more.
mathwonk said:I guess any time the exponential map is surjective one onto a compact manifold, could look at the shape of a "fundamental polygon" in the tangent space, that maps almost isomorphically onto the manifold, and it may become a question related to the billiard table problem of when a ball struck returns to its original position, or whether the course of the ball is closed or not. But the billiard table here is a non euclidean multidimensional polyhedron.
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Minimal geodesics are the shortest paths between two points on a curved surface, such as a sphere or a curved plane. They are analogous to straight lines on a flat surface.
Extending minimal geodesics indefinitely allows us to explore the entire surface and find new paths between points. It also helps us understand the overall curvature of the surface and its properties.
In theory, yes. However, in practice, it may not always be feasible due to factors such as the complexity of the surface, computational limitations, and physical constraints.
Extending minimal geodesics can have applications in fields such as mathematics, physics, and computer science. It can help us model complex surfaces, study the behavior of particles moving on curved surfaces, and develop algorithms for pathfinding and optimization problems.
There may be limitations in terms of accuracy and precision, as well as the computational resources required to extend the geodesics. Additionally, the surface may have regions where the geodesics cannot be extended due to singularities or other complexities.