Definition:Closed Geodesic

Definition

Let $\struct {M, g}$ be a connected Riemannian manifold.

Let $I = \closedint a b$ be a close real interval.

Let $\gamma : I \to M$ be a nonconstant geodesic segment.

Suppose:

$\map \gamma a = \map \gamma b$

$\map {\gamma'} a = \map {\gamma'} b$

where $\gamma'$ is the velocity of $\gamma$.

Then $\gamma$ is said to be a closed geodesic.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Completeness