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
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:
$i)$ $ \widetilde\varphi \left( {s\left( t \right)} \right) = s\left( t \right)\,\,\forall \,\,t \in \left[ {a,b} \right] $,
ii)be continuous
iii)$ $ L\left( {\widetilde\varphi _{\left| {\left[ {x,y} \right]} \right.} } \right) = \left| {x - y} \right|
* Try* Here I don´t know what I can do. Maybe consider the inverse function, of the s(t)$ function, and composing, but I´m not sure if the $s(t)$ has also an inverse ( it´s injective) . Anyway, if the inverse exist, how can I prove that this function is the function that I want? Sorry for this stupid question :/!!