Definition:Reparametrization of Admissible Curve

Definition

Let $M$ be a smooth manifold.

Let $\closedint a b, \closedint c d$ be closed real intervals.

Let $\gamma : \closedint a b \to M$ be an admissible curve.

Let $\phi : \closedint c d \to \closedint a b$ be a homeomorphism.

Let $\tuple {c_0, \ldots, c_k}$ be a finite subdivision of $\closedint c d$.

Suppose $\tuple {c_0, \ldots, c_k}$ is such that for all $i \in \N : 1 \le i \le k$ the restriction of $\phi$ to each subinterval $\closedint {c_{i - 1}} {c_i}$ is a diffeomorphism onto its image.

Then $\tilde \gamma := \gamma \circ \phi$ is called the reparametrization of the admissible curve $\gamma$, where $\circ$ denotes the composition of mappings $\gamma$ and $\phi$.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances