I have a manifold $M^n$ of dimension $n$ that I'm treating as embedded within an $n+1$ dimensional manifold $N^{n+1}$, $M^{n}\subset N^{n+1}$. Now I also have an $n-1$ dimensional manifold $\Sigma^(n-1)$ $\Sigma^{n-1}\subset M^{n}$. So we have $$\Sigma\subset M\subset N$$
To shake things up a bit up a bit let's say $\Sigma$ grows by a conformal factor $a(t)$ as we go along the normal to $ \Sigma $ in M. Then the inner product on $\Sigma$ changes accordingly, $$<,>_{\Sigma}\rightarrow a(t)^{2}<,>_{\Sigma}$$
I'm trying to find an expression for the unit normal vector to the hypersurface $M\subset N$ in terms of the conformal factor on $\Sigma$.
In case I'm not being clear let me give a simplistic example:
Consider the spaces $S^{1}\subset S^{1}\times\mathbb{R}\subset\mathbb{R}^{3}$. We have the circle embedded in the cylinder then our circle is conformally transformed as we move up or down the cylinder. What is the unit normal vector to $ S^{1}\times\mathbb{R}\ $?
If we choose cylindrical coordinates ($\rho,\phi,z$) aligned with our cylinder then the normal vector is infinitesimally:
$$\overrightarrow{n}=da(z)\hat{\rho}-dz\hat{z} $$
Such that our unit normal is: $$\hat{n}=\frac{\overrightarrow{n}}{|n|}=\frac{da}{\left(da^{2}+dz^{2}\right)^{1/2}}\hat{\rho}-\frac{dz}{\left(da^{2}+dz^{2}\right)^{1/2}}\hat{z}=\frac{da}{dz}\left(\left(\frac{da}{dz}\right)^{2}+1\right)^{-1/2}\hat{\rho}-\left(1+\left(\frac{da}{dz}\right)^{2}\right)^{-1/2}\hat{z}$$
Which can be noticably written as a unit complex vector: $$\hat{z}=e^{-i\theta}=cos(\theta)-isin(\theta)$$ (not to be confused with the former $\hat{z}$)
so that: $$\hat{n}^{\mu}\hat{n}_{\mu}=\hat{z}\bar{\hat{z}}=1$$
for some theta $$\theta=Cos^{-1}\left(\frac{da}{dz}\left(\left(\frac{da}{dz}\right)^{2}+1\right)^{-1/2}\right)$$ where the real and imaginary axis correspond to $\rho$ and $z$ respectively.
This setup makes sense as a check when I think about the normal bundle of $\Sigma$ in our general situation having $SO(2)$ fibers on $N$ (since it's always codimension 2).
I also know that the conformal group of an $(n-1)$ manifold is locally $SO(n,1)$, but I don't see how that comes into play here.
I believe I've done this construction right, and it would seem to apply to the general case as well. Is this correct?
How would I generalize this when the conformal factor (scale parameter) is a function of all of the coordinates $a(x(t)^i)$ on $M$? I'd expect the unit normal vector in $N$ necessarily involve a gradient of the scale factor then?. At the very least I'd like to find a coordinate invariant way of writing this expression in general.
For a little more background, I'm trying to find an expression for the second fundamental form of $M$ in $N$ in this situation, which turns out to be surprisingly complicated.