I have made an attempt at this exercise: Is the following alright?:
If we take ##\mathcal{E}=\mathbb{R}##, and ##\gamma:\mathcal{E}\rightarrow \mathbb{R}^2## defined by ## \gamma u= (x(au),au),## for ##u\in \mathcal{E}=\mathbb{R}##,
then ##\gamma_*u=a\frac{dx}{du}(au)\partial_{u^1}+a\partial_{u^2}##. This gives:
(a) ##du^2(\gamma_* u)=(a\frac{dx}{du}(au)\partial_{u^1}+a\partial_{u^2})(u^2)=0+a\cdot 1=a,##
(b) ##\gamma^1(u)=x(au)=x(\gamma^2 u)=(x\circ \gamma^2)## for all real ##u##, and
(c) ##g(\gamma_* u,\gamma_* u)=(a\frac{dx}{du}(au))^2-(a)^2=a^2[(\frac{dx}{du}(au))^2-1^2]=a^2\cdot 0=0.##
##\phantom{(c)}##Here we have used ##v=|\frac{dx}{du}|=1## everywhere.
I haven't yet tried showing uniqueness of ##\mathcal{E}## and ##\gamma##.