# Exterior Derivative on Curved Manifold (SpaceTime)

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.
• Oh Thanks! And what about if the connection $\nabla$ is not TorsionFree? The correct definition should be with the partial derivative $\partial$ then? – Andrea Di Pinto Jan 30 at 1:49
• Yes, if you mean the exterior derivative $\mathrm{d}$. – Qmechanic Jan 30 at 1:50
• Ok, perfect. And there exist some kind of "exterior covariant derivative" $\tilde{d}$ constructed with the connection $\nabla$? Or this has no use? – Andrea Di Pinto Jan 30 at 14:54