# Induced metric in submanifolds - What about the Christofells?

1. Jul 23, 2010

### tamiry

(EDIT fixed format)

Hi,
I have a question about properties which are common to a manifold and its submanifolds. I start with the metric.
$$M \subset N, dim(M) = m, dim (N) = m+1$$
let $$g^N$$ be the metric of N, so that $$(N,g^N)$$ is a riemanian manifold and N is a submanifold.
Now, I'm looking at N and I'm trying to understand what does $$g^M$$ looks like. WLOG I assume that in every point $$p \in M$$ there exists $$\phi$$ a homemorphism of a neighbourhood of p to $$U \subset R^{m+1}$$ $$p = \phi(U^1,...,U^m,U^{m+1} = 0)$$ I call the reduced $$\phi, \psi$$.
Now, I can see that
$$\partial \psi / \partial u^j = \partial \phi / \partial u^j$$ for $$1 \leq j \leq n$$ and that, $$\\ \partial \psi / \partial u^{m+1} = 0$$ (by definition) so I conclude that in U coordinates, $$g^N$$ has the form
$$\left(\begin{array}{cc}A_{m \times m}&*\\***&B_{1 \times 1}\end{array}\right)$$
This must be this way, of the inner product will not be induce correctly from N to M. A is exactly $$g^M$$
Now, I'm trying to check the christoffel symbols (so I could know what the covariant derivative is). I use the formula
$$\Gamma^k_{i j} = 1/2 * g^{k l} ( \partial g_{l j} / \partial u^i + ...)$$
And here is my problem. the factors in the brackets are identical for M and N, but I cant say the same about $$g^{k l}$$. If I could determine that * from above is zero (?) then I could say that the inverse of $$g^N$$ is
$$\left(\begin{array}{cc}A^{-1}&0\\C&D\end{array}\right)$$
but unfortunately, I dont know if I can choose coordinates, so that this property holds. Can I somehow make it happen? or is there another way to compute $$\Gamma^M$$ from $$\Gamma^N$$?

thanks

Last edited: Jul 23, 2010