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An alternative definition of the codifferential is given as formal adjoint of $d$ when this one is expressed using the covariant derivative:

$\delta\omega=-\frac 1 {(k-1)!}\nabla^i\omega_{i i_2 \ldots i_k}d x^{i_2}\wedge\ldots\wedge d x^{i_k}$

I found this formula in Pit-Mann Wong - Einstein Manifolds.

Eventually it turned out that this definition , with some adaptation, fits my needs.

I wish tot thank to Orbicular, but Johannes Ebert and Willie Wong for their help.

I would like to leave keep the question open for a while, because I am interested in other alternative answers.

I wish tot thank to Orbicular, Johannes Ebert and Willie Wong for their help.

1

One alternative definition of the codifferential is given as formal adjoint of $d$ when this one is expressed using the covariant derivative:

$\delta\omega=-\frac 1 {(k-1)!}\nabla^i\omega_{i i_2 \ldots i_k}d x^{i_2}\wedge\ldots\wedge d x^{i_k}$

I found this formula in Pit-Mann Wong - Einstein Manifolds.

Eventually it turned out that this definition, with some adaptation, fits my needs, but I would like to leave the question open, because I am interested in other alternative answers.

I wish tot thank to Orbicular, Johannes Ebert and Willie Wong for their help.