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I have some troubles understanding covariant derivatives in different basis.

Let's start by taking a manifold M with some metric g, then let's set a coordinate basis for the tangent bundle $\frac{\partial}{\partial x^i}$, the covariant derivative should be: $$\nabla_\mu V^\nu = \partial_\mu V^\nu + \Lambda^\nu_{\mu\rho}V^\rho$$ where $\Lambda$ are Christoffel symbols. What I don't understand is how to calculate: $$ div V = \nabla_\mu V^\mu = \ast d\ast (V) $$ where V in the last equation is the one form associated to the vector field V through musical isomorphism. Taking the above mentioned formula and doing all the calculations I obtain $$\ast d\ast(V) = \partial_\mu V^\mu$$ but I am missing all the Christoffel symbols. Why?

I have noticed that If I choose a non coordinate basis $e_a$, in particular an orthonormal one, then all Christoffel symbols are 0 and therefore this relation holds: $$div V = \nabla_a V^a = \partial_a V^a = \ast d\ast(V)$$ but how can I go back to the original coordinate basis to show the same relation in that basis? I am very confused about this topic, sorry if I wrote something completely wrong.

Thank you

Physics Koan
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    For starters, you’re neglecting the metric in the musical isomorphism. – Ted Shifrin Jun 19 '23 at 18:27
  • Using the differential forms formula you should easily arrive at the Voss-Weyl formula. Otherwise, it means you didn’t understand the Hodge star. Also, in an orthonormal frame the Christoffel symbols do not necessarily vanish. Also all the metric isomorphisms… – peek-a-boo Jun 19 '23 at 18:28
  • See here for a computation using the Hodge star, and see here for one using Lie derivatives, and here for a discussion using Christoffel symbols to arrive at the Voss-Weyl formula. – peek-a-boo Jun 19 '23 at 18:34
  • Thank you for the links, I am doing something wrong with the hodge star operator. In an orthonormal frame Christoffel symbols do not necessarily vanish because the metric is not guaranteed to be constant? – Physics Koan Jun 19 '23 at 19:21
  • the metric components in an orthonormal frame are of course constant; they’re $\delta_{ij}$. But, the Lie bracket of these vector fields doesn’t necessarily vanish, and that’s the issue. In a coordinate-induced orthonormal frame, they do commute, so the constancy of metric components coupled with vanishing commutator implies the Christoffel symbols vanish. – peek-a-boo Jun 19 '23 at 20:18

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