I'm trying to understand the Gibbons–Hawking term. There appears $K$- a trace of the second fundamental form, as they say.
As far as I know, the second fundamental form is the following thing: Suppose we have two manifolds $M$ and $N$, and $M$ is a surface in $N$ ($M \hookrightarrow N$). $g$ is a metrics on $N$, so $f^{*}g$ (pullback) is a metrics on $M$. Let $\rho$ to be a projection $T_{x}N \rightarrow T_{x}M$. Then $\nabla^M_{y}X=\rho(\nabla^N_{y}X)$ is a Levi-Civita connection on $M$ . Superscripts $M$ and $N$ are correspond to manifolds and $X$,$Y$ $\in \Gamma(TM)$. Then we have $\nabla^N_{y}X - \nabla^M_{y}X=(id-\rho)\nabla^N_{y}X=B(X,Y)$. $B(X,Y)$ is called a second fundamental form.
But all physicists I know write $K=h^{ab}K_{ab}=h^{ab}\nabla_{a}n_{b}$, where $h$ is induced metrics, and $n$ is a normal vector. (for example, check this link, slide number $5$, page $14$, where he introduced GH term for action. Or you can find other examples with ease by yourself).
So I don't understand how $n$ have appeared in definition of $K$. Can you explain it to me? Maybe I'm mistaken in my definition of second fundamental form?
I'm assuming $M$ is a hypersurface here, since you refer only to a single normal.
The first definition is the normal component of the derivative, and can be written $B(X,Y) = g(\nabla_Y X, n)$. (To match your definition strictly you must multiply this again by the vector field $n$; but since the normal bundle is one-dimensional this is useless information.)
The product rule lets us write this as $$B(X,Y) = \nabla_Y g(X,n) - g(X,\nabla_Y n),$$ but this first term vanishes because $X$ is tangent to $M$ while $n$ is orthogonal. Thus $B(X,Y) = -g(X,\nabla_Y n)$, or in index notation $B_{ij} = - \nabla_j n_i$. This is what you have up to a sign - some people choose the opposite sign convention so that they can simply write $B = \nabla n$.