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 $\gamma$.