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I am trying to calculate the entries of the Riemann curvature tensor $R^m_{\phantom{m}ijk}$ for the metric $g_{ij}$.

The Riemann-Christoffel tensor is given as \begin{align} R^m_{\phantom{m}ijk} = \frac{\partial}{\partial x^j}{m \brace i\,\,k} - \frac{\partial}{\partial x^k}{m \brace i\,\,j} + {n \brace i\,\,k}{m \brace n\,\,j} - {n \brace i\,\,j}{m \brace n\,\,k} \end{align} where the Christoffel symbol of second kind are given as \begin{align} {m\brace b\,\,c} = {m\brace c\,\,b} = g^{mb} \left[ac,b\right], \end{align} and the Christoffel symbol of first kind \begin{align} [ac, b] &= [ca,b] = \frac{1}{2} \left[\frac{\partial g_{ab}}{\partial x^c} + \frac{\partial g_{bc}}{\partial x^a} - \frac{\partial g_{ac}}{\partial x^b}\right]. \end{align}

I have to basically calculate all these symbols and insert them in the Riemann curvature tensor. Is there any way I can easily keep track of the elements of this fourth-order tensor ? (Visual tricks, or simply using simplifications to end up with trivial results.) The metric $g_{ij}$ is diagonal. So obviously there will be many trivial terms.

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    $\begingroup$ First, ignore the assumption that the metric is diagonal. It doesn't help. $\endgroup$
    – Deane Yang
    Oct 2 '15 at 12:57
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    $\begingroup$ Second, why are you doing this? $\endgroup$
    – Deane Yang
    Oct 2 '15 at 12:58
  • $\begingroup$ I agree with Deane, except if the metric is conformal to the euclidean one, in which case the formulas get much simpler. $\endgroup$ Oct 2 '15 at 12:58
  • $\begingroup$ I intend to contract the fourth order tensor and find the Ricci tensor. After that I intend to perform another contraction on the Ricci tensor. I want to visualize the contracted quantity for a given metric. $\endgroup$
    – imranal
    Oct 2 '15 at 13:07
  • $\begingroup$ You want to compute the scalar curvature of your metric $g$ if I understand correctly. Still the question stands, what use would you have of the formula you will get for the scalar curvature ? $\endgroup$ Oct 2 '15 at 13:32
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I managed to find a post on Physics StackExchange :

https://physics.stackexchange.com/questions/179393/any-tips-on-evaluating-riemann-tensor

which details what I feared. The process it self is too tedious. The best solution is to use a symbolic package which can do all these calculations, like for example the python package SymPy.

Another answer (from the same thread) is based on some thing called "Cartan formalism" (it is also the answer which has been up-voted the most).

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If it is something you can do with a computer do it. I can suggest you to use Sage Manifold which is already installed on Cloud SageMath To quickly know how to use you can look for Christoffel in this example Off course this is not the only option as you can use Mathematica and Maple. But this is Open Source, free to use and you don't have to install anything in your computer.

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