# Curves and geodesics

This is a very long problem of homework.

Definitions: We start by defining a curve as a continuous function $\phi :\left[ {a,b} \right] \to \left( {M,d} \right)$ where M is a metric space with metric d. We define the length of the curve $\phi :\left[ {a,b} \right] \to M$ as $$L\left( \varphi \right) = \mathop {\sup }\limits_{p \in P} \sum\limits_{k = 1}^n {d\left( {\varphi \left( {p_{k - 1} } \right),\varphi \left( {p_k } \right)} \right)}$$ where p runs over all the partitions P of $[a,b]$ i.e a finite collection of points, of the form $a = p_0 < p_1 < ... < p_n = b$ ( if not exist we just simply say that $L\left( \varphi \right) = \infty$ ) .

First part of the Problem:

$i)$ Let $\varphi$ be a curve that has finite length $L(\varphi)$ . Prove that there exist a function $s:[a,b] \to [0,L(\varphi)]$ such that $s\left( t \right) = L\left( {\varphi \left| {_{\left[ {a,t} \right]} } \right.} \right)$. Prove that $s$ it´s non decreasing, continuous and surjective.

Solution to the first part (Not complete)

I proved that $s$ it´s non decreasing. I also proved that if I have a partition P, and I add a new point to the partition, then the sum over that new partition it´s $\leqslant$ than the original. Using that $$L\left( {\varphi \left| {_{\left[ {a,t + \varepsilon } \right]} } \right.} \right) \leqslant L\left( {\varphi \left| {_{\left[ {a,t} \right]} } \right.} \right) + L\left( {\varphi \left| {_{\left[ {t,t + \varepsilon } \right]} } \right.} \right)$$. So to prove continuity it´s enough to prove that $$\mathop {\lim }\limits_{x \to 0^ + } L\left( {\varphi \left| {_{\left[ {j,j + x} \right]} } \right.} \right) = 0$$ i.e given any $\varepsilon > 0$ there exist a $\delta >0$ such that if $0<x<\delta$ then $L\left( {\varphi \left| {_{\left[ {j,j + x} \right]} } \right.} \right) < \varepsilon$
What I did here it´s trying to use the continuity of $\varphi$ but wasn´t work. For example choosing $\delta$>0 such that $|x-j|<\delta$ $\Rightarrow$ $d\left( {\varphi \left( x \right),\varphi \left( \delta \right)} \right) < \frac{\varepsilon } {2}$ So with this given a partition with "n" elements of the interval $\left[ {j,j + \delta } \right]$ , we know that $\sum\limits_{n\,sums} {d\left( {\varphi \left( {p_k } \right),\varphi \left( {p_{k - 1} } \right)} \right)} < n\varepsilon$ But that was all that I can do :/!!! I need help with this. This is not the problem. Someone has a book about metric geometry? ( involving geodesics , and others) .

Part 2 and final of the problem

Prove that there exist a function $\widetilde\varphi :\left[ {0,L\left( \varphi \right)} \right]: \to X$ such that: