Definition:Smooth Path/Simple/Real Cartesian Space

Definition

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a smooth path in $\R^n$.

$\rho$ is a simple smooth path (in $\R^n$) if and only if:

$(1): \quad \rho$ is injective on the half-open interval $\left[{a \,.\,.\, b}\right)$

$(2): \quad \forall t \in \left({a \,.\,.\, b}\right): \rho \left({t}\right) \ne \rho \left({b}\right)$

That is, if and only if $t_1, t_2 \in \left({a \,.\,.\, b}\right)$ with $t_1 \ne t_2$, then $\gamma \left({a}\right) \ne \gamma \left({t_1}\right) \ne \gamma \left({t_2}\right) \ne \gamma \left({b}\right)$.

Also see

Compare with the topological definition of an arc which requires that $\rho$ is injective on the closed interval $\left[{a \,.\,.\, b}\right]$.

A simple smooth path $\rho$ is injective on $\left[{ a \,.\,.\, b}\right]$ if and only if $\rho$ is not closed.