Can one use a coordinate transformation to show equivalence of line elements? Background
In my general relativity class the lecturer gave us three set of line elements:
$$ ds^2 = dx^2 + dy^2 $$ 
$$ ds^2 = dx'^2 + x'^2 dy'^2 $$
$$ ds^2 = dA^2 + \sin^2 (A) dB^2$$ 
And (later) told us the first two were related by coordinated transformations (Cartesian and polar co-ordinates). The third one describes a different geometry and thus cannot be related by a coordinate transformation. He then said it was an open problem for number of dimensions greater than $4$ regrading if  two arbitrary line-elements could be changed into another by a coordinate transformation
Questions
What is the name of this open problem? How is it solved in $4$ dimensions or less?     
 A: This problem, and variants of it, are known as (local) equivalence problems in the geometry literature. A mathematician would phrase your question as follows: given an open set $U$ on a $d$-dimensional manifold, and two metrics $g$, $h$ on $U$, then are $g$ and $h$ are the same up to a coordinate change?
In general this is a pretty difficult problem, as you can convince yourself. The main result is due to Cartan, who proved that $g$ and $h$ are the same if their Riemann tensors $R_{\mu \nu \rho \sigma}$ and its first $\frac{1}{2}d(d+1)$ covariant derivatives agree. The result of Cartan was later sharpened by Brans and Karlhede, and the resulting method of checking whether metrics are the same is known as the Cartan-Karlhede algorithm (meaning that you can teach it to a computer). Using this algorithm you still need to compute the first 7 derivatives of the Riemann tensor working in four dimensions, so you see that it's not an easy exercise to say the least. However, if your manifold is symmetric the complexity becomes drastically lower.
