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Consider the unit 2-sphere with its canonical metric $g_{\theta \theta} = 1$, $g_{\phi \phi} = \sin(\theta)$ and $g_{\theta \phi} = g_{\phi \theta} = 0$, the associated Levi-Civita connection has Christoffel symbols : $\Gamma^{\theta}_{\phi \phi} = -\cos(\theta)\sin(\theta)$, $\Gamma^{\phi}_{\theta \phi} = \Gamma^{\phi}_{\phi \theta} = \cot(\theta)$. I want to show that the volume form $\omega = \sin(\theta)\mathrm{d} \theta \wedge \mathrm{d} \phi$ is preserved under parallel transport in order to show that the covariant derivative preserves orientation. I use : $$\nabla_j \omega_{\alpha \beta} = \frac{\partial \omega_{\alpha\beta}}{\partial x^j} - \Gamma^\delta_{\alpha j}\omega_{\delta \beta} - \Gamma^\delta_{\beta j}\omega_{\delta \alpha}$$ and I find that all derivatives are zero except for : $$\nabla_{\theta}\omega_{\theta \phi} = 2\cos(\theta) $$ I can not find my error. Or is what I am trying to prove wrong ? If it is the case, then how do I prove that parallel transport preserves orientation ?

Julien
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  • I am not sure to understand. Indeed there is a minus in front of the Christoffel symbols in the expression of the covariant derivative, whereas for tangent vectors there is a plus, but I think I respected this rule when I wrote the expression of the covariant derivative of the volume form. Are you talking about this or something else ? Thanks. – Julien Nov 27 '19 at 19:11
  • Actually I think I found my mistake : I wrote : $$\nabla_j \omega_{\alpha \beta} = \frac{\partial \omega_{\alpha\beta}}{\partial x^j} - \Gamma^\delta_{\alpha j}\omega_{\delta \beta} - \Gamma^\delta_{\beta j}\omega_{\delta \alpha}$$ instead of : $$\nabla_j \omega_{\alpha \beta} = \frac{\partial \omega_{\alpha\beta}}{\partial x^j} - \Gamma^\delta_{\alpha j}\omega_{\delta \beta} - \Gamma^\delta_{\beta j}\omega_{\alpha \delta}$$ : in the last term the 2 indices of $\omega$ are reversed. I don't know if this is what you meant but thanks a lot anyway, it helped me find my mistake. – Julien Nov 28 '19 at 13:09

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