Riemannian Manifolds: Extending Geodesics Indefinitely

Thread starter Dragonfall
Start date Jun 19, 2013
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In summary, a Riemannian manifold is a mathematical concept used to describe curved spaces and surfaces. Extending geodesics indefinitely means finding a way to continually extend the shortest paths on the manifold without constraints or boundaries. This has various applications in mathematics and physics, such as describing the curvature of spacetime and analyzing shapes. Geodesics are a fundamental concept in Riemannian geometry and play a crucial role in understanding the geometry of the space. However, there are limitations to extending geodesics indefinitely, such as the complexity of the manifold and the presence of singularities.

Jun 19, 2013

Dragonfall

Given a closed Riemannian manifold, a point P on it and a nonzero vector V in its tangent space, can you extend a geodesic in that direction of V indefinitely? I count looping back onto itself as "indefinitely".

The theorem I have in my book only guarantees that this is possible locally near P.

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Jun 19, 2013

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1. What is a Riemannian manifold?

A Riemannian manifold is a mathematical concept that describes a space in which each point has a specific curvature. It is a generalization of the Euclidean space and is used to study curved surfaces and spaces in mathematics and physics.

2. What does it mean to extend geodesics indefinitely on a Riemannian manifold?

Geodesics are the shortest paths between two points on a curved surface. On a Riemannian manifold, extending geodesics indefinitely means finding a way to continually extend these paths without any constraints or boundaries. This allows us to study the behavior of objects moving freely on the manifold.

3. What are some applications of studying Riemannian manifolds and extending geodesics indefinitely?

Riemannian manifolds and extending geodesics indefinitely have numerous applications in mathematics and physics. They are used in general relativity to describe the curvature of spacetime, in computer vision and image processing to study and analyze shapes, and in optimization problems to find the shortest paths between two points.

4. How is the concept of geodesics related to Riemannian manifolds?

Geodesics are a fundamental concept in Riemannian geometry, which is the study of Riemannian manifolds. Geodesics are the paths that follow the direction of the greatest change in the curvature of the manifold, and they play a crucial role in understanding the geometry of the space.

5. Are there any limitations to extending geodesics indefinitely on a Riemannian manifold?

While the concept of extending geodesics indefinitely allows us to study the behavior of objects on a Riemannian manifold without any constraints, in reality, there are limitations. These limitations may arise due to the complexity of the manifold or the presence of singularities, which are points where the curvature becomes infinite. Additionally, the process of extending geodesics indefinitely can become computationally intensive for high-dimensional manifolds.