Definition:Forward Reparametrization of Curve
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Let $M$ be a smooth manifold.
Let $I, I' \subseteq \R$ be real intervals.
Let $\gamma : I \to M$ be a smooth curve.
Let $\phi : I' \to I$ be a diffeomorphism.
Let $\tilde \gamma$ be a curve defined by:
- $\tilde \gamma := \gamma \circ \phi : I' \to M$
where $\circ$ denotes the composition of mappings $\gamma$ and $\phi$.
Suppose $\phi$ is increasing.
Then $\tilde \gamma$ is called the forward reparametrization of $\gamma$.
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances