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Let $S$ be a $(n-1)$-submanifold of a $n$-manifold $M$ and that be a submanifold of $(n+1)$-manifold V. (All of the manifolds are assumed orientable) Let $V$ carry a metric of signature $(1,n)$. Using terminology as in Einstein's Theory let $n$ be a future directed unit normal to $M$ and $m$ be an unit normal to $S$ pointing outward which is tangent to $M$.

Let $h$ be a metric on $S$ , $g'$ be a metric on $M$ and $g$ be a metric on $V$ such that

$g_{\mu \nu}' = g_{\mu \nu} + n_\mu n_\nu$

and

$h_{\mu \nu} = g_{\mu \nu} + n_\mu n_\nu - m_\mu m _\nu$

(In each of the cases $\mu$ and $\nu$ run over the dimensions of the lowest dimensional manifold whose metric is involved)

I would like to know as to what is the motivation for the above choice of metrics that is often made. What makes the above "natural"?

To pull-back a metric on a submanifold, one needs to fix an embedding first. Here what is the implicit embedding?

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    @Ronaldo You can see use of this metric in Section 5 (Trapped Surface) of Chapter 13 (Singularities) of Yvonne's book on Einstein's Equations. Its between Page 414 to 4172010-09-22

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