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Consider the redefinition $$d\zeta_1=ad\theta,~d\zeta_2=a\sin\theta d\phi$$ so that the distance between two points on the surface of a sphere of radius $a$ becomes $$dl^2=a^2(d\theta^2+\sin^2\theta d\phi^2)=d\zeta_1^2+d\zeta_2^2=\delta_{ij}d\zeta_id\zeta_j$$ Can the first equation be called a coordinate transformation? Have I really transformed the line element on the surface of a sphere to Euclidean metric? But I'm told that transforming the metric tensor to $\delta_{ij}$ via a coordinate transformation is possible only for a flat surface.

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  • $\begingroup$ If you really did find a coordinate transformation from $θ,φ$ to $ζ_1, ζ_2$, why are you failing to display it? $\endgroup$ – Cosmas Zachos Mar 22 '19 at 21:25
  • $\begingroup$ @CosmasZachos In the first equation I displayed it. Why is this not a coordinate transformation is my question. Because they are not finite transformations? $\endgroup$ – mithusengupta123 Mar 23 '19 at 0:34
  • $\begingroup$ A coordinate transformation is a map of variables, right? $\endgroup$ – Cosmas Zachos Mar 23 '19 at 1:54
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No, your redefinition, by itself, does not specify a coordinate transformation.

Assume you did have a coordinate transformation. Look at your conjectural $\zeta_2(\theta,\phi)$, and work out its $$ d\zeta_2= \frac{\partial\zeta_2}{\partial\theta}d\theta + \frac{\partial\zeta_2}{\partial\phi}d\phi ~. $$ Compare with your peculiar ad hoc identification, $$ \frac{\partial\zeta_2}{\partial\theta}=0, \qquad \frac{\partial\zeta_2}{\partial\phi}=a \sin \theta. $$ The first equation is solved by all functions of only $\phi$ , but the second by $a\phi \sin \theta + b(\theta)$, so there is no such $\zeta_2$.

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