Let $\Gamma_{ij}^k$ be the Christoffel symbols of a time-varying metric $g$ on $\Sigma$, a 3-manifold (say). Let $k_{ij}$ be the second fundamental form of $\Sigma$ as embedded in $M = \mathbb{R} \times \Sigma$ with the Lorentzian metric $h = -dt^2 + g(t)$, normalized so that $\partial_t g_{ij} = -2k_{ij}$.
One frequently used definition is the "time-derivative of the connection" $\dot\Gamma_{ij}^k$, defined to be the tensor field $$ \dot\Gamma_{ij}^r = -(\nabla_i {k}_j^{ \ r} + \nabla_j {k}_j^{\ r} - \nabla^r k_{ij}) $$ where $\nabla$ is the connection on $\Sigma$. Question: Is this the same thing as $$ \frac{\partial}{\partial t}\Gamma_{ij}^r $$ in any coordinate system "adapted to $\Sigma$", i.e. $(x^i, i = 1, 2, 3)$ are coordinates on $\Sigma$ and $t$ is the time coordinate, and such that $h(\partial_t, \partial_i) = 0$ (so that the metric $h$ has the same form as given above)?
When I try to verify this, I get that $\frac{\partial}{\partial t} \Gamma_{ij}^r$ is equal to $\dot\Gamma_{ij}^k$ plus some mess involving Christoffel symbols that doesn't seem to obviously vanish. So I'm not sure if this is a well-known fact that, say, this is only tensorial in certain coordinates.