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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:

[tex]\Gamma^x_{xx}=\frac{1}{x}[/tex] and [tex]\Gamma^y_{yy}=\frac{2}{y}[/tex]

knowing that: [tex]R^\alpha_{\beta\gamma\delta}=\partial_\gamma \Gamma^\alpha_{\delta\beta}-\partial_\delta \Gamma^\alpha_{\gamma\beta}+\Gamma^\epsilon_{\delta\beta}\Gamma^\alpha_{\gamma\epsilon}-\Gamma^\epsilon_{\gamma\beta}\Gamma^\alpha_{\delta\epsilon}[/tex]

The result I have obtained is that

MY PROCEDURE HAS BEEN:

the only plausible non-zero components of the Riemann curvature tensor are:

[tex]R^\alpha_{\beta\gamma\delta}=\partial_\gamma \Gamma^\alpha_{\delta\beta}-\partial_\delta \Gamma^\alpha_{\gamma\beta}+\Gamma^x_{\delta\beta}\Gamma^\alpha_{\gamma x}+\Gamma^y_{\delta\beta}\Gamma^\alpha_{\delta y}-\Gamma^x_{\gamma\beta}\Gamma^\alpha_{\delta x}-\Gamma^y_{\gamma\beta}\Gamma^\alpha_{\delta y}[/tex]

[tex]\alpha=x\quad:\quad R^x_{\beta\gamma\delta}=\partial_\gamma \Gamma^x_{\delta\beta}-\partial_\delta \Gamma^x_{\gamma\beta}+\Gamma^x_{\delta\beta}\Gamma^x_{\gamma x}+\Gamma^y_{\delta\beta}\Gamma^x_{\delta y}-\Gamma^x_{\gamma\beta}\Gamma^x_{\delta x}-\Gamma^y_{\gamma\beta}\Gamma^x_{\delta y}[/tex]

[tex]\text{ }\quad\quad \Longrightarrow \quad R^x_{xxx}=\partial_x \Gamma^x_{xx}-\partial_x \Gamma^x_{xx}+\Gamma^x_{xx}\Gamma^x_{xx}+\Gamma^y_{xx}\Gamma^x_{xy}-\Gamma^x_{xx}\Gamma^x_{xx}-\Gamma^y_{xx}\Gamma^x_{xy}=0[/tex]

[tex]\alpha=y\quad:\quad R^y_{\beta\gamma\delta}=\partial_\gamma \Gamma^y_{\delta\beta}-\partial_\delta \Gamma^y_{\gamma\beta}+\Gamma^x_{\delta\beta}\Gamma^y_{\gamma x}+\Gamma^y_{\delta\beta}\Gamma^y_{\delta y}-\Gamma^x_{\gamma\beta}\Gamma^y_{\delta x}-\Gamma^y_{\gamma\beta}\Gamma^y_{\delta y}[/tex]

[tex]\text{ }\quad\quad \Longrightarrow \quad R^y_{yyy}=\partial_y \Gamma^y_{yy}-\partial_y \Gamma^y_{yy}+\Gamma^y_{yy}\Gamma^y_{yx}+\Gamma^y_{yy}\Gamma^y_{yy}-\Gamma^y_{yy}\Gamma^y_{yx}-\Gamma^y_{yy}\Gamma^x_{yy}=0[/tex]

Therefore, all the components of the Riemann Tensor are zero.

Thanks!!!

[tex]\Gamma^x_{xx}=\frac{1}{x}[/tex] and [tex]\Gamma^y_{yy}=\frac{2}{y}[/tex]

knowing that: [tex]R^\alpha_{\beta\gamma\delta}=\partial_\gamma \Gamma^\alpha_{\delta\beta}-\partial_\delta \Gamma^\alpha_{\gamma\beta}+\Gamma^\epsilon_{\delta\beta}\Gamma^\alpha_{\gamma\epsilon}-\Gamma^\epsilon_{\gamma\beta}\Gamma^\alpha_{\delta\epsilon}[/tex]

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?MY PROCEDURE HAS BEEN:

the only plausible non-zero components of the Riemann curvature tensor are:

[tex]R^\alpha_{\beta\gamma\delta}=\partial_\gamma \Gamma^\alpha_{\delta\beta}-\partial_\delta \Gamma^\alpha_{\gamma\beta}+\Gamma^x_{\delta\beta}\Gamma^\alpha_{\gamma x}+\Gamma^y_{\delta\beta}\Gamma^\alpha_{\delta y}-\Gamma^x_{\gamma\beta}\Gamma^\alpha_{\delta x}-\Gamma^y_{\gamma\beta}\Gamma^\alpha_{\delta y}[/tex]

[tex]\alpha=x\quad:\quad R^x_{\beta\gamma\delta}=\partial_\gamma \Gamma^x_{\delta\beta}-\partial_\delta \Gamma^x_{\gamma\beta}+\Gamma^x_{\delta\beta}\Gamma^x_{\gamma x}+\Gamma^y_{\delta\beta}\Gamma^x_{\delta y}-\Gamma^x_{\gamma\beta}\Gamma^x_{\delta x}-\Gamma^y_{\gamma\beta}\Gamma^x_{\delta y}[/tex]

[tex]\text{ }\quad\quad \Longrightarrow \quad R^x_{xxx}=\partial_x \Gamma^x_{xx}-\partial_x \Gamma^x_{xx}+\Gamma^x_{xx}\Gamma^x_{xx}+\Gamma^y_{xx}\Gamma^x_{xy}-\Gamma^x_{xx}\Gamma^x_{xx}-\Gamma^y_{xx}\Gamma^x_{xy}=0[/tex]

[tex]\alpha=y\quad:\quad R^y_{\beta\gamma\delta}=\partial_\gamma \Gamma^y_{\delta\beta}-\partial_\delta \Gamma^y_{\gamma\beta}+\Gamma^x_{\delta\beta}\Gamma^y_{\gamma x}+\Gamma^y_{\delta\beta}\Gamma^y_{\delta y}-\Gamma^x_{\gamma\beta}\Gamma^y_{\delta x}-\Gamma^y_{\gamma\beta}\Gamma^y_{\delta y}[/tex]

[tex]\text{ }\quad\quad \Longrightarrow \quad R^y_{yyy}=\partial_y \Gamma^y_{yy}-\partial_y \Gamma^y_{yy}+\Gamma^y_{yy}\Gamma^y_{yx}+\Gamma^y_{yy}\Gamma^y_{yy}-\Gamma^y_{yy}\Gamma^y_{yx}-\Gamma^y_{yy}\Gamma^x_{yy}=0[/tex]

Therefore, all the components of the Riemann Tensor are zero.

Thanks!!!

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