Let $(\widetilde{\mathcal{M}},\widetilde{g})$ be some pseudo-Riemannian manifold and $(\mathcal{M},g)$ a Riemannian submanifold of co-dimension $1$, i.e. a hypersurface. Then, the fundamental form is the symmetric $2$-tensor field defined by
$$k(X,Y):=\langle N,\widetilde{\nabla}_{X}Y\rangle$$
for all $X,Y\in\mathfrak{X}(\mathcal{M})$, where $\widetilde{\nabla}$ is the Levi-Civita connection of the ambient manifold $\widetilde{\mathcal{M}}$, $N$ the unit normal vector and $X,Y$ are arbitrarely extended to whole $\widetilde{\mathcal{M}}$ (the choice of extension does not matter). Since $N$ is a normal vector, this can also be written as
$$k(X,Y):=-\langle \widetilde{\nabla}_{X}N,Y\rangle$$
In physics texts, I have seen that in coordinates $k_{\mu\nu}=-N_{\mu;\nu}=-\widetilde{\nabla}_{\nu}N_{\mu}$ and I would like to derive this equality.
My attempt:
Let us use that $k$ is symmetric. Then:
$$k_{\mu\nu}=k(\partial_{\mu},\partial_{\nu})=k(\partial_{\nu},\partial_{\mu})=-\langle\widetilde{\nabla}_{\partial_{\nu}}(N^{\alpha}\partial_{\alpha}),\partial_{\mu}\rangle=-N^{\alpha}\langle\widetilde{\nabla}_{\partial_{\nu}}\partial_{\alpha},\partial_{\mu}\rangle-\partial_{\nu}N^{\alpha}\langle\partial_{\alpha},\partial_{\mu}\rangle=\\=-N^{\alpha}\widetilde{\Gamma}^{\beta}_{\alpha\nu}\langle\partial_{\beta},\partial_{\mu}\rangle-\partial_{\nu}N^{\alpha}g_{\alpha\mu}=-\partial_{\nu}N_{\mu}-\widetilde{\Gamma}^{\beta}_{\alpha\nu}g_{\beta\mu}N^{\alpha},$$
which is seems to be different from
$$-N_{\mu;\nu}=-\widetilde{\nabla}_{\nu}N_{\mu}=-\partial_{\nu}N_{\mu}+\widetilde{\Gamma}_{\mu\nu}^{\alpha}N_{\alpha}.$$
Is there something wrong in my calculation? Or are the two expressions somehow equivalent?
Best Answer
The mistake is $\partial_\nu N^\alpha g_{\alpha\mu} = \partial_\nu N_\mu$. This does not hold, because $\partial_\nu N^\alpha$ is not a tensor (i.e., it does not transform like a tensor when you change coordinates). Instead, you have to do the following: \begin{align*} \partial_\nu N^\alpha g_{\alpha\mu} &= \partial_\nu(N^\alpha g_{\alpha\mu}) - N^\alpha\partial_\nu g_{\alpha\mu}\\ &= \partial_\nu N_{\mu} - N^\alpha(\Gamma_{\nu\alpha}^\beta g_{\beta\mu} +\Gamma_{\nu\mu}^\beta g_{\beta\alpha}) \end{align*}
It's also worth noting that the identity can also be proved using what I like to call "differentiation by parts" as follows: \begin{align*} k(X,Y) &= \langle N,\widetilde\nabla_XY\rangle\\ &= \partial_X\langle N,Y\rangle - \langle \widetilde\nabla_XN,Y\rangle\\ &= - \langle \widetilde\nabla_XN,Y\rangle, \end{align*} because $\langle N,Y\rangle = 0$.