# First fundamental form in the Gibbons-Hawking-York boundary term

Let me expose my problem, I am trying to perform the explicit variation of the Gibbons-Hawking-York boundary term, $$S_{GH}=\int_{\partial M} d^{n-1}x\sqrt{\left|h\right|}K$$ The problem I have is that in the calculation of $\delta\sqrt{\left|h\right|}$, it seems like I should carry the calculation as if $h$ was the first fundamental form $$h_{\mu\nu}=g_{\mu\nu} - \sigma n_{\mu} n_{\nu}$$ inasmuch as I obtain the good result doing so. First, I use the identity $$\delta\sqrt{\left|h\right|} = -\frac12 \sqrt{\left|h\right|} h_{\mu\nu} \delta h^{\mu\nu}$$ then I express $\delta h$ in fuction of $\delta g$ $$\delta h^{\mu\nu} = \delta g^{\mu\nu} -\sigma \delta n^\mu n^\nu -\sigma n^\mu\delta n^\nu$$ and using the fact that $h_{\mu\nu} n^\mu = 0$, we obtain

$$\delta\sqrt{\left|h\right|} = -\frac12 \sqrt{\left|h\right|} h_{\mu\nu} \delta g^{\mu\nu}.$$

My problem is that $h$ is not the first fundamental form in the first expression, but the induced metric. The determinant of the first fundamental form is 0 in gaussian normal coordinates, so it seems like I am skipping a step in this derivation, but I just cannot find what (I never had a course in differential geometry, so my understanding of the difference between the first fundamental form and the induced metric is really poor).

I got the answer reading a book from E. Poisson, what I was doing was indeed wrong, you have to start with the induced metric given by $$h_{ab}= g_{\mu\nu}e^{\mu}_a e^{\nu}_b$$ where $$e^{\mu}_a=\frac{\partial x^{\mu}}{\partial y^a}$$ are the tangent vectors to curves of the hypersurface. Then, you just replace $g$ by $h$ in the usual relation $$\delta\sqrt{\left|h\right|} = -\frac12 \sqrt{\left|h\right|} h_{ab} \delta h^{ab}.$$ Using the Kronecker invariance, one finds
$$\delta\sqrt{\left|h\right|} = \frac12 \sqrt{\left|h\right|} h^{ab} \delta h_{ab}.$$
Then using the fact that $$\delta h_{ab} =\delta g_{\mu\nu}e^\mu_a e^\nu_b$$ since the tangent vectors are invariant, we finally get $$\delta\sqrt{\left|h\right|} = \frac12 \sqrt{\left|h\right|} h^{\mu\nu} \delta g_{\mu\nu}$$ where we have used the definition of the PROJECTOR $$h^{\mu\nu} = h^{ab}e^\mu_a e^\nu_b.$$ And this is where the problem was coming, $h^{\mu\nu}$ is NOT the induced metric, but a projector associated to that induced metric.
and your coordinates $h_{\mu \nu}=g_{\mu \nu}-\sigma n_{\mu}n_{\nu}$ arent gaussian coordinates in general.
gaussian are for example flrw metrics like $ds^2=-dt^2+h_{ij}(t)dx^idx^j$ with spacelike $i,j$ where there is no mixed time/space basis term in the second part of the line element, only a scalefactor dependent on $t$