Prove that torsion is invariant by isometries and reparametrizations. Prove that torsion is invariant by isometries and reparametrizations.
How to prove it? please help me. 
if f is a isometry in $R^3$, $f\begin{pmatrix} x\\ y \\ z \end{pmatrix}$ = M$\begin{pmatrix} x \\ y \\ z \\\end{pmatrix}$ +v, $M^tM =I_3$ 
where v is a fixed vector in $R^3$
 A: The torsion of a curve is given by
$$ \frac {(r'\times r'').r'''}{( r' \times r'').( r' \times r'') }.$$
where $r(t)$ is your parametrisation and the primes denote differentiating w.r.t. $t$.
Your isometry acts by sending $r \mapsto Mr$. It also sends $r' \mapsto Mr'$, $r'' \mapsto Mr''$ and $r''' \mapsto Mr'''$. So all you have to do is to substitute the new expressions into the formula for the torsion. Be aware that for any vectors $u$ and $v$,and for any orthogonal mapping $M$, it is true that $(Mu)\times(Mv) = M(u\times v)$ and $(Mu).(Mv) = u.v$, and since $M$ is an isometry, $M^TM$ is the identity matrix. Thus
$$ (r'\times r'').r''' \mapsto (Mr'\times Mr'').Mr''' = M(r'\times r'').Mr''' '= (r'\times r'').r''' $$
and similarly,
$$( r' \times r'').( r' \times r'') \mapsto (M r' \times M r'').( M r' \times M r'') = M( r' \times r'').M( r' \times r'')\\ = ( r' \times r'').M^TM( r' \times r'') = ( r' \times r'').( r' \times r'')$$
For reparametrisations, suppose $\tau$ is your new parameter. Differentiate using the chain rule and the product rule:
$$ \frac{dr}{d\tau} = \frac{dr}{dt}\frac{dt}{d\tau},$$
$$ \frac{d^2r}{d\tau^2} = \frac{d^2r}{dt^2}\left(\frac{dt}{d\tau}\right)^2 + \frac{dr}{dt}\frac{d^2t}{d\tau^2},$$
$$ \frac{d^3r}{d\tau^3} = \frac{d^3r}{dt^3}\left(\frac{dt}{d\tau}\right)^3 + 2 \frac{d^2r}{dt^2}\left(\frac{dt}{d\tau}\right)\left(\frac{d^2t}{d\tau^2}\right)+\frac{dr}{dt}\frac{d^3t}{d\tau^3},$$
and substitute these expressions into your formula. Remember this fact about the triple product: $(u\times v).w$ is zero if any two of $u,v,w$ are equal. Thus
$$ \left(\frac{dr}{d\tau} \times \frac{d^2 r}{d\tau^2} \right).\frac{d^3 r}{d\tau^3} = \left( \frac{dt}{d\tau} \right)^6 \left(\frac{dr}{dt} \times \frac{d^2 r}{dt^2} \right).\frac{d^3 r}{dt^3}$$
and
$$ \left(\frac{dr}{d\tau} \times \frac{d^2 r}{d\tau^2} \right).\left(\frac{dr}{d\tau} \times \frac{d^2 r}{d\tau^2} \right) =  \left( \frac{dt}{d\tau} \right)^6 \left(\frac{dr}{dt} \times \frac{d^2 r}{dt^2} \right) .  \left(\frac{dr}{dt} \times \frac{d^2 r}{dt^2} \right) $$
and hence
$$
\frac{\left(\frac{dr}{d\tau} \times \frac{d^2 r}{d\tau^2} \right).\frac{d^3 r}{d\tau^3}}{ \left(\frac{dr}{d\tau} \times \frac{d^2 r}{d\tau^2} \right).\left(\frac{dr}{d\tau} \times \frac{d^2 r}{d\tau^2} \right)} = \frac{\left(\frac{dr}{dt} \times \frac{d^2 r}{dt^2} \right).\frac{d^3 r}{dt^3}}{ \left(\frac{dr}{dt} \times \frac{d^2 r}{dt^2} \right) .  \left(\frac{dr}{dt} \times \frac{d^2 r}{dt^2} \right)}
$$
