Riemann Tensor knowing Christoffel symbols (check my result)

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I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are:



$Gamma^x_xx=frac1x$ and $Gamma^y_yy=frac2y$



knowing that: $R^alpha_betagammadelta=partial_gamma Gamma^alpha_deltabeta-partial_delta Gamma^alpha_gammabeta+Gamma^epsilon_deltabetaGamma^alpha_gammaepsilon-Gamma^epsilon_gammabetaGamma^alpha_deltaepsilon$



The result I have obtained is that all the components of the Riemann curvature tensor are zero. Is this correct? If it is, what does it mean that all the components are zero?



Thanks!!!




differential-geometry riemannian-geometry tensors




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    I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are:



    $Gamma^x_xx=frac1x$ and $Gamma^y_yy=frac2y$



    knowing that: $R^alpha_betagammadelta=partial_gamma Gamma^alpha_deltabeta-partial_delta Gamma^alpha_gammabeta+Gamma^epsilon_deltabetaGamma^alpha_gammaepsilon-Gamma^epsilon_gammabetaGamma^alpha_deltaepsilon$



    The result I have obtained is that all the components of the Riemann curvature tensor are zero. Is this correct? If it is, what does it mean that all the components are zero?



    Thanks!!!




    differential-geometry riemannian-geometry tensors




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are:



      $Gamma^x_xx=frac1x$ and $Gamma^y_yy=frac2y$



      knowing that: $R^alpha_betagammadelta=partial_gamma Gamma^alpha_deltabeta-partial_delta Gamma^alpha_gammabeta+Gamma^epsilon_deltabetaGamma^alpha_gammaepsilon-Gamma^epsilon_gammabetaGamma^alpha_deltaepsilon$



      The result I have obtained is that all the components of the Riemann curvature tensor are zero. Is this correct? If it is, what does it mean that all the components are zero?



      Thanks!!!




      differential-geometry riemannian-geometry tensors




      share|cite|improve this question











      I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are:



      $Gamma^x_xx=frac1x$ and $Gamma^y_yy=frac2y$



      knowing that: $R^alpha_betagammadelta=partial_gamma Gamma^alpha_deltabeta-partial_delta Gamma^alpha_gammabeta+Gamma^epsilon_deltabetaGamma^alpha_gammaepsilon-Gamma^epsilon_gammabetaGamma^alpha_deltaepsilon$



      The result I have obtained is that all the components of the Riemann curvature tensor are zero. Is this correct? If it is, what does it mean that all the components are zero?



      Thanks!!!





      differential-geometry riemannian-geometry tensors





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









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