Considering the 2-Form Gauge Potential $B_{\mu \nu}$, we can write $dB=H$.

In a flat manifold we have that $H_{\mu \nu \rho}=\partial_{\mu}B_{\nu \rho}+\partial_{\rho}B_{\mu \nu}+\partial_{\nu}B_{\rho \mu}$, does this still applies on a curved manifold (spacetime)? Or we have to substitute the normal derivatives with the covariant derivatives? ($H_{\mu \nu \rho}=\nabla_{\mu}B_{\nu \rho}+\nabla_{\rho}B_{\mu \nu}+\nabla_{\nu}B_{\rho \mu}$).

I know that if the potential is a 1-form ($A_\mu$) the result is the same, but I want to know what is the correct definition for a generic $p$-form potential.


The exterior derivative $\mathrm{d}$ needs no connection $\nabla$. The two types of derivatives on $p$-forms are the same if the connection $\nabla$ is torsionfree.

  • $\begingroup$ Oh Thanks! And what about if the connection $\nabla$ is not TorsionFree? The correct definition should be with the partial derivative $\partial$ then? $\endgroup$ – Andrea Di Pinto Jan 30 at 1:49
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    $\begingroup$ Yes, if you mean the exterior derivative $\mathrm{d}$. $\endgroup$ – Qmechanic Jan 30 at 1:50
  • $\begingroup$ Ok, perfect. And there exist some kind of "exterior covariant derivative" $\tilde{d}$ constructed with the connection $\nabla$? Or this has no use? $\endgroup$ – Andrea Di Pinto Jan 30 at 14:54
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    $\begingroup$ Yes, it's mathematically well-defined. Whether it is relevant in physics is another matter. $\endgroup$ – Qmechanic Jan 30 at 17:07

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