Riemannian Geometry/Parametrisation of curves

Definition (regular curve):

Let $(M,g)$ be a Riemannian manifold. A curve $\gamma :I\to M$ is called regular if and only if for all $t\in {\overset {\circ }{I}}$, we have

$\gamma '(t)\neq 0$.

Proposition (existence of parametrisation by arc-length of regular curves):

Let $\gamma :I\to \mathbb {R} ^{n}$ be a regular, continuously differentiable curve, where $I=(a,b)\subseteq \mathbb {R}$ is an open, connected interval. Then there exists a different open, connected interval $J=(c,d)\subseteq \mathbb {R}$ and a continuously differentiable bijective function $\rho :J\to I$, such that

$\forall t\in J:\|(\gamma \circ \rho )'(t)\|=1$.

Proof: By the chain rule, we have

$(\gamma \circ \rho )'(t)=\gamma '(\rho (t))\rho '(t)$

for all $t\in J$. Hence, the equation $\|(\gamma \circ \rho )'(t)\|=1$ becomes equivalent to

$|\rho '(t)|={\frac {1}{\|\gamma '(\rho (t))\|}}$,

and the existence of a solution to the latter equation implies the existence of a solution to the former. Moreover, it suffices to solve

$\rho '(t)={\frac {1}{\|\gamma '(\rho (t))\|}}$,

because the right hand side will always be positive. From Peano's theorem, we may infer the existence of a $\rho$ satisfying the given identity on a neighbourhood of $0$, where we impose the initial condition $\rho (0)=t_{0}$, where $t_{0}\in J$ is arbitrary. Then we may extend the solution to the maximum interval, $\Box$

Proposition (existence of regular modification):

Let $\gamma :[0,1]\to \mathbb {R} ^{n}$ be a curve which is twice continuously differentiable. Then there exists a modified curve ${\tilde {\gamma }}:[0,c]\to \mathbb {R} ^{n}$, defined on an interval $[0,c]$, where $c>0$ is some positive number, such that

$\forall t\in [0,c]:\|{\tilde {\gamma }}'(t)\|=1$.

Proof: Define

$U:=\{t\in [0,1]|\gamma '(t)\neq 0\}=(\gamma ')^{-1}(\mathbb {R} ^{n}\setminus \{0\})$;

by continuity of $\gamma '$, this is an open subset of $[0,1]$. It decomposes into at most countably many connected components

$\bigcup _{n\in \mathbb {N} }U_{n}$, where $U_{n}=(a_{n},b_{n})$ for all $n\in \mathbb {N}$;

when there are only finitely many connected components, say $N$ many, we will set $a_{n}=b_{n}=1$ for $n>N$ by convention.

Now fix an $n$ such that $a_{n}<b_{n}$. We would like to modify $\gamma |_{(a_{n},b_{n})}$ such that for all points $t$ in the domain of the modification (whatever that may be), the absolute value of the derivative of the modification equals $1$. By induction on $n$, we assume that we have already found a modification ${\tilde {\gamma }}_{n-1}:[0,c_{n}]\to \mathbb {R} ^{n}$ that traverses $\Box$

Definition (parametrisation by arc-length):

Proposition (characterisation of parametrisation by arc-length):