= Solution
Take $n^an_a=-1$. The <first fundamental form>, or <induced metric>, is the <spatial projection tensor>
$$
\boxed{h_{ab}=g_{ab}+n_an_b},\qquad h_a{}^b=\delta_a{}^b+n_an^b.
$$
With the convention used in the question, define the <second fundamental form> by
$$
\boxed{K_{ab}=-h_a{}^ch_b{}^d\nabla_cn_d}.
$$
Locally $n_a$ is proportional to the gradient of a defining function for the <hypersurface>, so $h_a{}^ch_b{}^d\nabla_{[c}n_{d]}=0$ and $K_{ab}=K_{ba}$. Both derivative indices are projected tangentially. If two extensions of $n$ agree on $\Sigma$, their difference vanishes there, as does its tangential derivative, so $K_{ab}$ is independent of the extension.
Back to article page
