# A particular method of pulling back a metric on a submanifold

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?

{Now the LaTeX is compiling}

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@Anirbit: You needed some curly braces. There does appear to be a bug with the g' in the first formula; I had to move the prime after the subscripts to get the formula rendered. Please check that I have interpreted your subscripting correctly! (It doesn't look right to me, but I changed as little as possible. How does the unit normal vector $m$ get two subscripts?) (BTW, the double use of $n$ as a dimension and a unit normal is risky.) – whuber Sep 17 '10 at 14:56
@Whuber Thanks for your efforts. But none of the formulas are still readable. :) I can't figure out whats going on. I definitely did not want to put two indices on any vector! I wonder if some weird font rendering is happening. But I can't see anything rendered! – Anirbit Sep 17 '10 at 17:19
Can you tell from which book/what page you got this sentence? The embedding must be the inclusion map, since M is a subset of V /S is a subset of M. – Ronaldo Sep 19 '10 at 22:53
@Ronaldo Which sentence are you referring to? – Anirbit Sep 21 '10 at 13:30
Actually, the whole question. Since you are asking why is this a "good choice", I assume you read it somewhere, maybe in a GR book. I would like to know the reference to see if there is any hidden hypothesis about the coordinate systems used (it seems to be a coordinate system adapted to the embedded submanifold), and what is the context. Try to write g_{\mu\nu} = g'_{\mu\nu} - n_\mu n_\nu, with g'_{\mu,n+1}=0 and let \mu,\nu run from 1 to n+1. See if this helps in some way. – Ronaldo Sep 22 '10 at 1:05