Rectifiable Curves

Subjects: Metric and Normed Spaces
Links: Rectifiable Curves in Rn, Continuity on Metric Spaces

A path in $X$ is a continuous function $\sigma :[a,b]\to X$. The length of $\sigma$ is defined as

or it can be denoted as, meaning the same thing

we get that

We call it rectifiable.

We see that $\mathfrak{L}:{\mathcal{C}}^0([a,b],X)\to \mathbb{R}\cup \{\mathrm{\infty }\}$ is lower semicontinuous, but it is not continuous.

Let $\rho :[\alpha ,\beta ]\to [a,b]$ be a continuous, nondecreasing surjective function and $\sigma \in {\mathcal{C}}^0([a,b],X)$ be a path from $x$ to $y$ in $X$, then the new path $\sigma \circ \rho \in {\mathcal{C}}^0([\alpha ,\beta ],X)$ is also a path from $x$ to $y$ in $X$. It is called a ********************reparemetrization of $\sigma$.

If $\sigma \circ \rho$ is a reparamitraztion of $\sigma$ , then $\mathfrak{L}(\sigma )=\mathfrak{L}(\sigma \circ \rho )$

Let’s conside the subspsace of the paths from $x$ to $y$ meaning

with the uniform metric

Let $x,y\in X$, such that $x\ne y$, and $c\in \mathbb{R}$. Then if

then ${\mathfrak{L}}^{\le c}$ is not compact

We say that a rectifiable path $\sigma \in {\mathcal{C}}^0([0,1],X)$ is parametrized proportionally by arc length if

Given a rectifiable path $\sigma \in {\mathcal{C}}^0([0,1],X)$, we define the function $\lambda :[0,1]\to [0,\mathfrak{L}(\sigma )]$ as

is a function that is continuous, non decreasing and surjective

We will denote the space of special paths from $x$ to $y$ in $X$, that are parametrized proportionally by arc length as

For each rectifiable $\sigma \in {\mathcal{T}}_{x,y}(X)$ then there exists $\hat{\sigma }\in {\hat{\mathcal{T}}}_{x,y}(X)$ such that $\mathfrak{L}(\sigma )=\mathfrak{L}(\hat{\sigma })$

Existence of Geodesic Paths (Minimum Length Paths)

Let $X$ be a compact metric space and $x,y\in X$. If there exists a path from $x$ to $y$ in $X$, then there exists a path with minimum lenght path from $x$ to $y$ in $X$