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I came across the following definition:

The Sobolev space $W^{k,p}_1(M)$ is the space of differential forms $\alpha\in\Omega^pM$ such that $$\|\alpha\|^2_0=\int_M\alpha\wedge \star\alpha<\infty \qquad \&\qquad \|\nabla^l\alpha\|^2_0<\infty \quad 1\leq l\leq k,$$ i.e. $\alpha$ is in $L^2$ together with its covariant derivatives up to order $k$.

I was wondering how the covariant derivative of any $p$-form is defined. I guess it is constructed from the Levi-Civita connection on the bundle $\Lambda^pTM^*$ but I don't get how to apply it to the previous definition. Don't we need a direction $X\in \mathfrak{X}(M)$ for the covariant derivatives? Are we checking the previous condition on every possible direction?

Watanabe
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    Does this answer your question? Covariant derivative for higher rank tensors In particular, the answer gives a formula of a covariant derivative $\nabla_X T$ of a $(p, q)$-tensor. – Arctic Char Sep 09 '21 at 19:35
  • And we represent $\nabla T$ as a $(p, q+1)$ tensor by $\nabla T(X, \cdots ) = (\nabla _XT) (\cdots)$. – Arctic Char Sep 09 '21 at 19:37
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    The definition you give only makes sense to differential forms, but if you use the equivalent definition$$|T|_0^2:=\int_M\langle T,T\rangle dV_g$$then in makes sense for all tensors. (Here $\langle\ ,\ \rangle$ denotes the inner product on tensors of any type induced by $g$.) – Kajelad Sep 09 '21 at 20:02

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