- #1
tasguitar7
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Consider two pseudo-Riemmannian manifolds, ##M## and ##N##. Suppose that in coordinates ##y^\mu## on ##M## and ##x^\mu## on ##N##, the Riemann curvatures ##R^M## and ##R^N## of ##M## and ##N## are related by a coordinate transformation ##y = y(x)##:
\begin{equation*}
R^N_{\rho\mu\sigma\nu} = R^M_{\alpha\beta\gamma\lambda}\frac{\partial y^\alpha}{\partial x^\rho}\frac{\partial y^\beta}{\partial x^\mu}\frac{\partial y^\gamma}{\partial x^\sigma}\frac{\partial y^\lambda}{\partial x^\nu}.
\end{equation*}
This is intended to mean that the curvatures are related everywhere by coordinate transformation, although some care may need to be taken with respect to this condition when changing charts in the atlas.
Anyway, if two manifolds have such a relationship everywhere between their curvatures, does that imply that their metrics are related by coordinate transformation:
\begin{equation*}
g^N_{\mu\nu} = g^M_{\alpha\beta}\frac{\partial y^\alpha}{\partial x^\mu}\frac{\partial y^\beta}{\partial x^\nu}?
\end{equation*}
If so or if not, how can you show it? The question is essentially a generalized version of that an everywhere vanishing Riemann curvature implies flatness.
\begin{equation*}
R^N_{\rho\mu\sigma\nu} = R^M_{\alpha\beta\gamma\lambda}\frac{\partial y^\alpha}{\partial x^\rho}\frac{\partial y^\beta}{\partial x^\mu}\frac{\partial y^\gamma}{\partial x^\sigma}\frac{\partial y^\lambda}{\partial x^\nu}.
\end{equation*}
This is intended to mean that the curvatures are related everywhere by coordinate transformation, although some care may need to be taken with respect to this condition when changing charts in the atlas.
Anyway, if two manifolds have such a relationship everywhere between their curvatures, does that imply that their metrics are related by coordinate transformation:
\begin{equation*}
g^N_{\mu\nu} = g^M_{\alpha\beta}\frac{\partial y^\alpha}{\partial x^\mu}\frac{\partial y^\beta}{\partial x^\nu}?
\end{equation*}
If so or if not, how can you show it? The question is essentially a generalized version of that an everywhere vanishing Riemann curvature implies flatness.
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