Consider a Riemannian 3-manifold $M$ with boundary $\partial M$ with unit normal $n$, and mean curvature $H$. Consider the scalar curvature $R$. I'm interested in computing
$n^a \nabla_a R = \langle \nabla R, n \rangle$ at $\partial M$.
I'm wondering if there is some formula for this in terms of the mean curvature of the boundary, and I thought it best to ask here before trying to derive the formula. My great hope is that in the case $H=0$, we obtain $n^a \nabla_a R =0 $.
I have no idea if this is true, but it does turn out to be true in the case of the standard three dimensional slices of Schwarzchild space-time (trivially) and also of Reissner-Nordstrom (less trivially), so there is some hope.
Any help is appreciated, and thanks in advance.
There is a hope here, but a few more things need to be taken into the consideration. I don't know the exact answer, so let me just to contribute my twopence to the discussion.
In a slightly more general situation, where $M$ is an $n$-dimensional manifold with a Riemannian metric $g$, let us denote by $\nabla$ the Levi-Civita connection for the metric $g$, and $Riem$. $Ric$, and $R$ be the corresponding Riemannian, Ricci, and scalar curvatures. If $\overline{M} = \partial M$ is the boundary of $M$, and hence an embedded hypersurface, we have the induced Riemannian metric $\overline{g}$ on it, and the Levi-Civita connection of this metric, its Riemannian, Ricci, and scalar curvatures will be denoted as $\overline{\nabla}$, $\overline{Riem}$, $\overline{Ric}$, and $\overline{R}$, respectively. Let $N$ be a choice of the unit vector along the boundary. We also need the notation $L$ for the second fundamental form, and $H$ for the mean curvature. (Since we are talking about derivatives, our consideration can be though as purely local).
In the notation as given above, there is a well-known consequence of the Gauss equation: $$ \overline{R} = R - 2 Ric_{a b} N^a N^b + (n - 1)^2H^2 - |L|^2 $$ which holds along the hypersurface (boundary).
Now, the point is that the derivatives with respect to the ambient $\nabla$ do not make sense for most of the terms in this equation, unless you come up with a certain extension (slices, foliation, solutions of Cauchy equations... - ?).