Exterior Derivative on Curved Manifold (SpaceTime)

Question

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.

Answer 1

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.