Definition:Geodesic/Formal Definition
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Definition
Let $M$ be a smooth manifold with or without boundary.
Let $I \subseteq \R$ be a real interval.
Let $\gamma : I \to M$ be a smooth curve on $M$.
Let $\gamma'$ be the velocity of $\gamma$.
Let $\nabla$ be a connection on $M$.
Let $D_t$ be the covariant derivative along $\gamma$ with respect to $\nabla$.
Suppose:
- $\forall t \in I : D_t \gamma' = 0$.
Then $\gamma$ is called the geodesic (with respect to $\nabla$).
Examples
Sphere
A geodesic on the surface $S$ of a sphere between points $A$ and $B$ on $S$ is the part of the great circle on which $A$ and $B$ both lie.
Also see
- Results about geodesics can be found here.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Covariant Derivatives Along Curves