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Let us denote a 1 form on manifold M with $\eta$ which in a chart looks like $\eta=\eta_{\mu}dx^{\mu}$ where $\eta_{\mu}$ are smooth functions on M. Now given the coordinate vector fields $\frac{\partial}{\partial x^{\mu}}$, $$\nabla_{\nu}\eta\equiv\nabla_{\frac{\partial}{\partial x^{\nu}}}\eta$$ is a (0,1) tensor field, i.e, another 1 form. so, it makes sense to talk about $(\nabla_{\nu}\eta)_{\mu}$ and I can see that (after acting $\nabla_{\nu}\eta$ on a coordinate vector field) the following holds $$(\nabla_{\nu}\eta)_{\mu}=\nabla_{\nu}\eta_{\mu}-\Gamma^{\rho}_{\mu\nu}\eta_{\rho}.$$ Note that since $\eta_{\mu}$ are smooth functions on M, $\nabla_{\nu}\eta_{\mu}=\eta_{\mu,\nu}$ are ordinary partial derivatives of $\eta_{\mu}$. The above equation can be rewritten as: $$\eta_{\mu;\nu}=\eta_{\mu,\nu}-\Gamma^{\rho}_{\mu\nu}\eta_{\rho}.$$ My question is why in the physics/general relativity literature then $\nabla_{\nu}\eta_{\mu}$ denotes the covariant derivative of the 1 form $\eta$ along the coordinate vector field $\frac{\partial}{\partial x^{\nu}}$? The covariant derivative of $\eta$ along $\frac{\partial}{\partial x^{\nu}}$, denoted by $\nabla_{\nu}\eta$ is a (0,1) tensor field whose components are denoted by $(\nabla_{\nu}\eta)_{\mu}$ (the left hand side of the second equation above) where as $\nabla_{\nu}\eta_{\mu}$ are mere partial derivatives of the component functions $\eta_{\mu}$.

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    $\begingroup$ That's just abuse of notation. When they write $\nabla_\nu\eta_\mu$, they mean $(\nabla_\nu\eta)_\mu$. $\endgroup$ – MBN Oct 14 '17 at 7:34
  • $\begingroup$ I guess, then, they denote $(\nabla_{\nu}X)^\mu$ by $\nabla_{\nu}X^{\mu}$ for vector field $X=X^{\mu}\frac{\partial}{\partial x^{\mu}}$ as well which to me is a serious abuse of notation and may lead to terrible confusions. $\endgroup$ – user78032 Oct 14 '17 at 12:36
  • $\begingroup$ Yes, they do. (two more characters) $\endgroup$ – MBN Oct 14 '17 at 19:29
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Usually $\nabla_\mu$ is not taken to be an operator by itself. Or, you can take it to be an operator, with the understanding that certain indices are tensor indices and certain indices are counting indices, and when $\nabla_\mu$ acts on something with tensor indices, it means "take the abstract covariant derivative, then take the components".

So $\nabla_\mu A^\nu$ is shorthand notation for $(\nabla A)_{\mu}^{\ \nu}$.

A convention that is probably not widespread, but what I personally use (if I have to mix Ricci calculus with modern differential geometry) is to use two, separate notation for covariant derivatives. The $\nabla_X Y$ notation for invariant notation and $\nabla_\mu A^\nu$ for component notation. In this sense, $$ \nabla_\mu A^\nu=\partial_\mu A^\nu +\Gamma^\nu_{\mu\sigma}A^\sigma $$ but $$ \nabla_{\partial_\mu}A^\nu=\partial_\mu A^\nu. $$

The two dont mix in the sense that I'll never write something as $$ \nabla_\mu A $$ where $ A=A_\mu dx^\mu $, and I'll never write something as $\nabla_X A^\mu$, unless I actually mean $X(A^\mu)=X^\nu\partial_\nu A^\mu$.

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