# Curvature Scalar in 2-d

Where should I start from to show that curvature scalar (RiemannScalar) is

2$\frac{R_1212}{det (g_μ√)}$

?

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I am asked to show it by the symmetries of the Riemann tensor by the way.

Ok here is my thoughts, I try to stay away from the connection coefficients. So, I don't write the R tensors in form of $\Gamma$'s. So I am trying:

R= g$^{αβ}$ R$_{αβ}$
= g$^{αβ}$ R$^{c}$$_{αcβ}$
= g$^{αβ}$ g$^{αb}$ R$_{bαcβ}$

but now I can't have the R$_{αβαβ}$ form. Since it is 2-d, I put α=1 and β=2, but the c and b contractions doesn't give me what I want.

How can get that 2R$_{1212}$ ?

Last edited:
Bill_K
remember that gαβ is the inverse matrix of gαβ. And it's easy to take the inverse of a 2 x 2 matrix.

Write out completely the individual components of gαβ in terms of the gαβ components and see what you get.

pervect
Staff Emeritus
My thoughts are to start by looking at the only completely anti-symmetric 2-form , which must have components

$$\left[ \begin{array}{cc} 0 & R \\ -R & 0 \\ \end{array} \right]$$

Now we know that R_abcd = 0 if a=b or c=d by the anti-symmetry properties, and also that R_abcd = R_bacd and that R_abcd = -R_abdc

This, and a little thought, gives us the value of all the components of R, which can be described as an anti-symmetric 2d array of two-forms, i.e. it looks like the array above, but the members of the array are the anti-symmetric two-forms.

Next we just have to compute the contractions to get the Riemann tensor and scalar, which I'm too lazy to do by hand.

Thank you very much.

Yes I rewrited everything with g lower index.

I am not sure about that symmetry:

R$^{αβ}$$_{αβ}$= - R$^{βα}$$_{βα}$

But I feel I am close to it. Thank you again.