Closed geodesics

Definition

Closed geodesics are curves on a Riemannian manifold that are locally distance minimizing and return to their starting point, creating a loop. These paths represent critical points of the energy functional and play a vital role in understanding the geometric and topological properties of manifolds, often relating to concepts such as conjugate points, stability, and the overall structure of the space.

5 Must Know Facts For Your Next Test

1. Closed geodesics exist on compact Riemannian manifolds due to the minimization properties of the length functional.
2. In some cases, closed geodesics can be used to classify the topology of the underlying manifold, revealing important geometric insights.
3. The existence of closed geodesics is often proven using variational methods, involving critical point theory and Morse theory.
4. Every compact Riemannian manifold has at least one closed geodesic due to the results related to minimizing sequences and compactness arguments.
5. Closed geodesics can be unstable or stable depending on the nature of perturbations in their direction, which can be analyzed using conjugate points and Morse indices.

Review Questions

• How do closed geodesics relate to conjugate points on a Riemannian manifold?
  • Closed geodesics are closely tied to the concept of conjugate points because conjugate points indicate where a geodesic ceases to be a local minimizer. If you have a closed geodesic, and you find conjugate points along it, this tells you about the stability of that geodesic. The presence of conjugate points along a closed path suggests that small perturbations might lead to different paths, indicating a loss of local minimum status for length.
• Discuss how the Morse index theorem applies to closed geodesics and what implications this has for their stability.
  • The Morse index theorem plays an essential role in analyzing closed geodesics by determining how many directions allow for a decrease in energy at critical points. For a closed geodesic, its Morse index gives insight into its stability; if the index is low, it suggests more stability in that direction. This relationship helps us understand not just whether closed geodesics exist but also their nature—whether they are stable (attracting nearby paths) or unstable (repelling nearby paths).
• Evaluate the significance of Bonnet-Myers theorem in relation to closed geodesics on Riemannian manifolds.
  • The Bonnet-Myers theorem establishes that compact Riemannian manifolds with positive Ricci curvature are not only compact but also have finite fundamental groups. This theorem reinforces the idea that such manifolds must contain closed geodesics due to their geometric properties. In essence, it tells us that certain curvature conditions guarantee not just the existence of closed loops but also significant implications for their topological structure, aiding in classifying manifolds based on their closed geodesics.

Related terms

Riemannian manifold:

A smooth manifold equipped with a Riemannian metric, allowing for the measurement of lengths and angles, which is crucial in defining geodesics.

Conjugate points:

Points along a geodesic where the geodesic fails to be a local minimum for length, indicating a change in the stability of geodesics.

Morse index: The number of negative eigenvalues of the second variation of the energy functional at a critical point, indicating the number of directions in which one can perturb a closed geodesic.