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Induced metric in submanifolds - What about the Christofells?

  1. Jul 23, 2010 #1
    (EDIT fixed format)

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

    Last edited: Jul 23, 2010
  2. jcsd
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