Induced metric in submanifolds - What about the Christofells?

[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 can't 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 don't 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

 

[jvicens]

for your help

Hi there,

Thank you for your question. It is a very interesting one and it shows that you have a strong understanding of manifold and submanifold properties. To answer your question, let's first review some key concepts.

A Riemannian manifold is a manifold equipped with a metric tensor, which defines the inner product between tangent vectors at each point on the manifold. This metric tensor is usually denoted as g and it determines the distance between two points on the manifold. In your case, N is a submanifold of M, which means that N is embedded in M and inherits its metric from M. This means that the metric tensor of N, denoted as g^N, is a restriction of the metric tensor of M, denoted as g^M. In other words, g^N is a projection of g^M onto N.

Now, let's look at the Christoffel symbols. These symbols are related to the metric tensor and describe the connection between tangent vectors on a manifold. In your case, the Christoffel symbols of N, denoted as \Gamma^N, are a restriction of the Christoffel symbols of M, denoted as \Gamma^M. This means that \Gamma^N is also a projection of \Gamma^M onto N. Therefore, to compute \Gamma^M from \Gamma^N, you can use the projection properties of g^N and \Gamma^N onto N. This will give you the correct Christoffel symbols for M.

To address your concern about the coordinates, it is true that the choice of coordinates can affect the form of the metric tensor and the Christoffel symbols. However, in this case, since N is a submanifold of M, the coordinate system on N is a subset of the coordinate system on M. This means that the same coordinates can be used for both manifolds, and the projection properties of g^N and \Gamma^N will still hold.

I hope this helps clarify your question. Let me know if you have any further questions or if you need any additional clarification. Keep up the good work on your understanding of manifolds and submanifolds!