Riemann Curvature Tensor in 2D

[Woolyabyss]

Homework Statement

Show that the Riemann curvature tensor in 2d is given by $R_{abcd} =\frac{R}{2}(g_{ac}g_{bd} - g_{ad}g_{bc})$

Relevant Equations

$R = R_{ab} g^{ab}$ and $R_{ab} = R_{acb}^{c}$

Since in 2D the riemman curvature tensor has only one independent component, $R = R_{ab} g^{ab}$ can be reversed to get the riemmann curvature tensor.

Write
$R_{ab} = R g_{ab}$

Now
$R g_{ab} = R_{acbd} g^{cd}$
Rewrite this as
$R_{acbd} = Rg_{ab} g_{cd}$
My issue is I'm not sure how they caught a second term? Any help would be appreciated.

 

[robphy]

Now
$R g_{ab} = R_{acbd} g^{cd}$
Rewrite this as
$R_{acbd} = Rg_{ab} g_{cd}$
My issue is I'm not sure how they caught a second term? Any help would be appreciated.

Your candidate Riemann tensor doesn't have the expected index-symmetries.
Generally, you can't undo a contraction to arrive at your last line.
There may be other expressions that lead to the same contracted term.