A nonconstant geodesic whose position and tangent velocity repeat after a positive time. On a flat torus it is a straight line modulo the lattice, closing when its displacement is a Euclidean lattice vector. A primitive closed geodesic traverses its image without being an iterate of a shorter periodic geodesic.
A primitive closed geodesic is a closed geodesic with its smallest positive period, rather than a repeated traversal of a shorter periodic geodesic. A degree- lift of a primitive base geodesic is still primitive upstairs when its monodromy permutation has a cycle of exact length : no smaller number of traversals returns to the starting sheet.
Articles by others on the same topic
A **closed geodesic** is a type of curve on a manifold that has several important properties in differential geometry and topology. Here are the key characteristics: 1. **Geodesic**: A geodesic is a curve that locally minimizes distance and is a generalization of the concept of a "straight line" to curved spaces. It can be defined as a curve whose tangent vector is parallel transported along the curve itself.