I have been deriving multiple Riemann tensors for the past few days (in 2D for simplicity sake) and for some reason, no matter what functions I put in the vector of transformation properties, I always get all 0's for my Riemann tensors.(adsbygoogle = window.adsbygoogle || []).push({});

Note: By vector of transformation properties, I am referring to the vector R that you differentiate with respect to all the axes in order to derive the basis vectors that you dot product to derive the elements of your metric tensor. For example, the vector of transformation properties for 2D polar coordinates is R= <rcos(θ) , rsin(θ)>. This is what I mean.

I have tried all sorts of functions within my vector. I have tried:

R=<rθ , r/θ> where x^{1}= r and x^{2}= θ

I've tried:

R=<sin(r), cos(θ)>

I've tried:

R= <e^{r}, e^{θ}>

I have tried:

R= <sin(θ)cos(ø), sin(θ)sin(ø)>

etc...

No matter what, my Riemann tensor comes out all 0.

Is there a reason for that? Is it because I am choosing relatively simple functions and need to forsake simplicity if I want to see some curvature? Is there some property of the Riemann tensor that guarantees that it will be all 0 under certain conditions that I don't know about?

Just what kind of functions would it take for me to put inside of that vector for me to be able to derive a Riemann tensor with non zero elements?

I heard about this object called a 2-sphere that apparently yields curvature of a 2 dimensional manifold. What are the transformation properties that I could use to derive a spatial geometry like that of a 2-sphere?

It just seems very unlikely that it is a coincidence that I always seem to get 0 matrices for my Riemann tensors.

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# Is there any particular reason on why I keep getting flat space?

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