Is a Riemannian Metric Invariant Under Any Coordinate Transformation?

[AlephClo]

Q1: How do we prove that a Riemannian metric G (ex. on RxR) is invariant with respect to a change of coordinate, if all we have is G, and no coordinate transform?
G = ( x2 -x1 )
( -x1 x2 )

Q2: Since the distance ds has to be invariant, I understand that it has to be proved independantly of a specific coordinate transform. Any relationships between a given Riemannian metric and coordinate transforms?

Thank you

 

[andrewkirk]

The best approach to defining Riemannian metrics is coordinate-free, as a map that takes two vectors in the tangent space and returns a scalar. It then follows automatically that the metric is coordinate-invariant.

For your specific case, you need to show that, for an arbitrary 2x2 change of coordinate matrix L and vectors $u,v\in\mathbb{R}^2$:
$$u^TGv=u'^TG'v'$$
where the $T$ superscript denotes transpose and the 'primed' items are transformed to the new basis via $L$, that is:

$$u'=Lu,\ v'=Lv,\ G'=(L^{-1})^TGL^{-1}$$

 

[AlephClo]

Very clear. Thank you very much Andrewkirk