I have seen the post Explicit Variation of Gibbons-Hawking-York Boundary Term on variation of Gibbons-Hawking term, that was really helpful, however, I have problem evaluating $\delta K$ and getting exactly to :
$$\delta K= - \frac{1}{2} K^{\mu\nu} \delta g_{\mu\nu} - \frac{1}{2} n^{\mu}\left(\nabla^{\nu} \delta g_{\mu\nu} - g^{\nu\lambda} \nabla_{\mu} \delta g_{\nu\lambda} \right) + D_{\mu} c^{\mu}$$ where $ c_{\mu} = - \frac{1}{2} h_{\mu}{}^{\lambda} \delta g_{\nu\lambda} n^{\nu} ~.$
I've seen the webpage http://jacobi.luc.edu/Useful.html and could work out $$\delta K_{\mu\nu}=\frac{1}{2} n^\alpha n^\beta \delta g_{\alpha\beta}K_{\mu\nu}+\delta g_{\lambda\rho}n^\rho(n_\mu K^\lambda_{ \;\;\nu}+n_\nu K_\mu^{\;\;\lambda})-\dfrac{1}{2}h_\mu^{\;\;\lambda}h_\nu^{\;\;\rho}n^\alpha(\nabla_\lambda \delta g_{\alpha\rho}+\nabla_\rho \delta g_{\lambda\alpha}-\nabla_\alpha \delta g_{\lambda\rho})$$
I tried to evaluate $\delta K$ in the following manner : \begin{align} \delta K&=\delta(K_{\mu\nu}g^{\mu\nu})=-K^{\mu\nu}\delta g_{\mu\nu}+g^{\mu\nu}\delta K_{\mu\nu}\\ &=-K^{\mu\nu}\delta g_{\mu\nu}+\frac{1}{2}n^\alpha n^\beta \delta g_{\alpha\beta }K-\frac{1}{2}h^{\lambda\rho}n^\alpha (\nabla_\lambda \delta g_{\alpha\rho}+\nabla_\rho \delta g_{\lambda\alpha}-\nabla_\alpha \delta g_{\lambda\rho})\\ &=-K^{\mu\nu}\delta g_{\mu\nu}+\frac{1}{2}n^\alpha n^\beta \delta g_{\alpha\beta }K-\frac{1}{2}n^\alpha(2 \nabla^\rho \delta g_{\alpha\rho}-g^{\lambda\rho}\nabla_\alpha \delta g_{\lambda\rho})+\frac{1}{2}n^\lambda n^\rho n^\alpha \nabla_\lambda \delta g_{\alpha \rho}\\ &= - \frac{1}{2} K^{\mu\nu} \delta g_{\mu\nu} - \frac{1}{2} n^{\mu}\left(\nabla^{\nu} \delta g_{\mu\nu} - g^{\nu\lambda} \nabla_{\mu} \delta g_{\nu\lambda} \right)-\frac{1}{2}K^{\mu\nu}\delta g_{\mu\nu}+\frac{1}{2}n^\mu n^\nu \delta g_{\mu\nu} K-\frac{1}{2} n^\mu h^{\nu\lambda}\nabla_\lambda \delta g_{\mu\nu} \end{align} In the third line I've used $h^{\lambda\rho}=g^{\lambda\rho}-n^\lambda n^\rho$ If I want to reach the same answer I stated above for $\delta K$, the last three terms should equal $ D_{\mu} c^{\mu}$ Is my calculation right? What shoud I do next? I tried to expand $\nabla_\lambda \delta g_{\mu\nu}$ in terms of the induced metric, but the answer gets so messy and I can't reach the appropriate answer.
