Closed Geodesics

Definition

Closed geodesics are curves on a Riemannian manifold that are locally length-minimizing and return to their starting point, effectively forming a loop. They represent paths that connect back to themselves without self-intersection and play an essential role in understanding the geometric structure of manifolds, particularly in relation to completeness, the behavior of exponential maps, and the nature of conjugate and focal points.

5 Must Know Facts For Your Next Test

1. Closed geodesics exist in various contexts, including on compact Riemannian manifolds where they can often be guaranteed by the existence of certain curvature conditions.
2. The Hopf-Rinow theorem states that if a Riemannian manifold is complete, any two points can be connected by a geodesic, which supports the idea of closed geodesics in complete spaces.
3. Closed geodesics can be characterized as critical points of the energy functional associated with curves on the manifold, making them central in variational calculus.
4. In spaces with constant positive curvature, such as spheres, every closed curve can be approximated by closed geodesics.
5. The presence of closed geodesics can lead to interesting topological implications for the manifold, influencing its fundamental group and other invariants.

Review Questions

• How do closed geodesics relate to the concept of completeness in Riemannian geometry?
  • Closed geodesics highlight significant properties related to completeness since the Hopf-Rinow theorem asserts that if a manifold is complete, then every pair of points can be joined by geodesics. In complete manifolds, closed geodesics exist and serve as examples of paths that can loop back to their starting point. This interconnection between closed geodesics and completeness shows how geometric structure is preserved even when curves return to their origins.
• Discuss how the exponential map aids in understanding closed geodesics and their behavior within a manifold.
  • The exponential map is crucial for analyzing closed geodesics because it translates tangent vectors into actual geodesic curves. When considering a point on a Riemannian manifold, the exponential map allows one to visualize how small perturbations can form loops or closed paths. By observing how these loops behave under the exponential map, we gain insight into the global structure of the manifold and the nature of closed geodesics therein.
• Evaluate the significance of conjugate points concerning closed geodesics and their implications for variational principles.
  • Conjugate points are significant because they indicate where a geodesic ceases to be locally length-minimizing. When analyzing closed geodesics through variational principles, understanding where conjugate points occur along these paths is essential for determining stability and local minima. The relationship between closed geodesics and conjugate points sheds light on the curvature characteristics of the manifold and reveals important information about the geometry that could lead to broader insights regarding its topological structure.

Related terms

Geodesic:

A geodesic is a curve that locally minimizes distance, serving as the shortest path between points in a Riemannian manifold.

Conjugate Points:

Conjugate points are pairs of points along a geodesic where the geodesic ceases to be a local distance minimizer, indicating regions of curvature in the manifold.

Exponential Map:

The exponential map at a point on a Riemannian manifold translates tangent vectors into geodesics, effectively linking local linear structures with global geometric properties.