![](https://steemitimages.com/DQmTTskWT8sQvyFqDP1hfareWzet4kKygvWYdVjJABSBBsf/image.png)

In my previous posting on Riemann geometry [[math,computation] Riemann geometry: Ricci scalar written in metric explicitly](https://steemit.com/math/@beoped/math-computation-riemann-scalar-written-in-metric) we express Ricci scalar in terms of metric only. In this post we will express Ricci curvature in terms of metric only. 

Starting from the defintion of Ricci curvature

![](https://steemitimages.com/DQmQjFKTB7TDd44tDVNu8YzgBFVPV4ZLYUc2ja9hdcejXHK/image.png)

The interesting fact about Ricci-curvature is that it it symmetric under exchange of mu,nu. [We can prove this via the property of Riemann curvature tensor]

Now let us recall the expression of Riemannian connection

![](https://steemitimages.com/DQmewKkBSLG4yYUJmizPBMDt8Dw2Hhhokmukj4No4HNTC1j/image.png)

By plugging this to the formula and re-orgaanize we have

![](https://steemitimages.com/DQmVcBpJ8uJJVmMPHMj21Ej1mHwHpTzeYiKMeYnuTgKwkeW/image.png)

The reason why I didn't just expand the first line but write it as a second line is to show the symmetric properties of Ricci curvature explicitly. Note the expression on (,) notation means 
![](https://steemitimages.com/DQmUL6JH4yNL9YfRxMahJHjEEqZx8fxW85f8zG3RgPsYcbF/image.png)
here factor 1/2 means the normalization of two. 

Now let us back to the above long expression. For the first line to the second line we used following equation 

![](https://steemitimages.com/DQmXHGYTaooUycpQv1aJSYhJvtHyUGSHEUPGjhXc1R5nBMn/image.png)

For matching up colored terms we re-label the dummy index and use the symmetric properties of metric which contracted with. 


To check symmetric properties, let us decompose and see the results again


![](https://steemitimages.com/DQmSm4DA1mEq2eVpLRXpTzd6rJ9ixh6PkPhej8U5nFt2Cx6/image.png)


Except red terms we can easily see the symmetric properteis between exchange of mu,nu. 

The symmetric properties of red term can be revealed from this equation 
![](https://steemitimages.com/DQmbNxAq3AL8FSzURMmkVmiM8vDPvMejC1AqQhrrqHsfuAY/image.png)


Now we are done. 

And again by contraction with metric g^{\mu\nu}, the computation are same in  [[math,computation] Riemann geometry: Ricci scalar written in metric explicitly](https://steemit.com/math/@beoped/math-computation-riemann-scalar-written-in-metric)  and obtain the same Ricci scalar expression