Consider a globally-hyperbolic manifold $(M,g)$, i.e. $M\cong\mathbb{R}\times\Sigma$ with metric $g=-\beta\mathrm{d}t^{2}+h_{t}$ where $\beta\in C^{\infty}(M)$ and $h_{t}$ is a time-dependent Riemannian metric on $\Sigma$. Now, let $E$ be a vector bundle over $M$ and consider a connection $\nabla$. In some lecture notes, I have seen the symbol $$\nabla\vert_{\Sigma}$$ i.e. the connetion $\nabla$ "restriced to $\Sigma$" and I am not sure how to interpret it.
Let us make an example: Consider $E:=T^{\ast}M$ so that $\Gamma(E)=\Omega^{1}(M)$. Now, in coordinates, $A\in\Gamma(E)$ can be written as $A=A_{\nu}\mathrm{d}x^{\mu}$ and $\nabla A\in\Gamma^{\infty}(T^{\ast}M^{\otimes 2})$ the 2-tensor given by $$(\nabla A)_{\mu\nu}:=\nabla_{\mu}A_{\nu}:=\partial_{\mu}A_{\nu}-\Gamma_{\mu\nu}^{\lambda}A_{\lambda}$$ What does $\nabla\vert_{\Sigma}$ mean in this context and how does it act on coordinates? I would expect that it is the connection acting on $\Gamma(E\vert_{\Sigma})$ where $\Sigma$ has to be understood as $\{0\}\times\Sigma$. Is this correct?