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It's known (see here for example) that after a rescaling of the metric $\tilde{g}=e^{2\omega}g$, we can find a new connection $\tilde\nabla$ associated to the new metric:

$ \tilde\nabla _X Y = \nabla _X Y + (X \omega )Y + (Y \omega )X - g(X,Y) \operatorname{grad}\omega \tag{1}, $

where $\nabla$ is the Levi-Civita connection of $(M,g)$. In coordinates:

$ \tilde\Gamma^{k}_{ij}=\Gamma^{k}_{ij} + \delta_{i}^{k} \partial_j \omega + \delta_{j}^{k} \partial_i \omega - g_{i j} g^{k l} \partial_{l} \omega. \tag{2} $

My question is: is the new connection $\tilde\nabla$ compatible with the new metric $\tilde g$? I am using Equation (1) together with the property

$ X[\tilde g(Y,Z)] = (\tilde\nabla_X\tilde g)(Y,Z)+\tilde g(\tilde\nabla_XY,Z)+\tilde g(Y,\tilde\nabla_XZ)\tag{3}, $

but I am not getting $\tilde\nabla\tilde g = 0$. Instead $\tilde\nabla\tilde g$ is proportional to the new metric. Is this right?

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1 Answer 1

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The connection $\tilde{\nabla}$ should be the Levi-Civita connection associated to the metric $\tilde{g}$ and so it should be compatible with $\tilde{g}$. And indeed,

$$ \tilde{g}(\tilde{\nabla}_X Y, Z) = e^{2\omega} g(\nabla_X Y + (X\omega)Y +(Y\omega)X - g(X,Y) \operatorname{grad} \omega, Z) = \\ e^{2\omega} \left( g(\nabla_X Y, Z) + (X\omega)g(Y,Z) + (Y\omega) g(X,Z) - g(X,Y) g(\operatorname{grad} \omega, Z) \right) = \\ e^{2\omega} \left( g(\nabla_X Y, Z) + d\omega(X)g(Y,Z) + d\omega(Y) g(X,Z) - d\omega(Z) g(X,Y) \right). $$

Similarly,

$$ \tilde{g}(Y, \tilde{\nabla}_X Z) = e^{2\omega} \left( g(\nabla_X Z, Y) + d\omega(X) g(Y,Z) + d\omega(Z) g(X,Y) - d\omega(Y) g(X,Z) \right).$$

Finally,

$$ (\tilde{\nabla}_X \tilde{g})(Y,Z) = X\tilde{g}(Y,Z) - \tilde{g}(\tilde{\nabla}_XY, Z) - \tilde{g}(Y,\tilde{\nabla}_X Z) = \\ 2e^{2\omega} d\omega(X)g(Y,Z) + e^{2\omega} \left( Xg(Y,Z) - (\left( g(\nabla_X Y, Z) + d\omega(X)g(Y,Z) + d\omega(Y) g(X,Z) - d\omega(Z) g(X,Y) \right) - \left( g(\nabla_X Z, Y) + d\omega(X) g(Y,Z) + d\omega(Z) g(X,Y) - d\omega(Y) g(X,Z) \right) \right) = \\ e^{2\omega} \left( (\nabla_X g)(Y,Z) + 2d\omega(X)g(Y,Z) + d\omega(Z)g(X,Y) + d\omega(Y)g(X,Z) - d\omega(X)g(Y,Z) - d\omega(Y)g(X,Z) - d\omega(X)g(Y,Z) -d\omega(Z)g(X,Y) \right) = e^{2\omega} (\nabla_X g)(Y,Z) = 0. $$

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