1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Levi-Civita connection and Christoffel symbols

  1. Jan 31, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that [tex]g(d \sigma ^k, \sigma _p \wedge \sigma _q) = \Gamma _{ipq} - \Gamma _{iqp}[/tex]


    2. Relevant equations
    Given $$\omega_{ij}=\hat e_i \cdot d \hat e _j = \Gamma_{ijk} \sigma^k$$, we can also say that $$d \hat e_j = \omega^i_j \hat e_i$$. Where $$\sigma^k, \sigma_p, \sigma_q$$ are to be one forms.

    3. The attempt at a solution
    I'm self -studying this material, and I just want to make sure I'm making the right connections!

    So what I did was to say, $$g(d\sigma^k, \sigma_p \wedge \sigma_q) =g(\omega_{ji} \sigma^j, \sigma_p \wedge \sigma_q) = g(\Gamma_{jik} \sigma^k \wedge \sigma^j, \sigma_p \wedge \sigma_q) = g(-\Gamma_{ijk} \sigma^k \wedge \sigma^j, \sigma_p \wedge \sigma_q)$$

    $$= -\Gamma_{ijk}g(\sigma^k \wedge \sigma^j, \sigma_p \wedge \sigma_q) = -\Gamma_{ijk}[\delta^k_p \delta^j_q - \delta^k_q \delta^j_p] = -\Gamma_{ijk}\delta^k_p \delta^j_q + \Gamma_{ijk} \delta^k_q \delta^j_p = -\Gamma_{iqp}+\Gamma_{ipq} = \Gamma_{ipq} - \Gamma_{iqp}$$

    Cool, that's what we wanted, but I'm not sure if everything I did was valid, so I'm just seeing if it's fine what I did.
    My justification for saying $$d\sigma^k = \omega_{ji} \sigma^j $$ is that we see that $$d \hat e_j = \omega^i_j \hat e_i$$ since $$\hat e_j $$is a basis vector, and $$\sigma_i$$ is a one form that can also form a basis, they should have the same connection. Thus, $$d\sigma_i = \omega^k_i \sigma_k = \omega^k_i g_{kj} \sigma^j = \omega_{ji} \sigma^j$$

    We know $$\Gamma_{jik} = - \Gamma_{ijk}$$ by metric compatibility.

    Sorry for the long post, just need some pointers, or reassurance that this is all right! (I've posted this on another forum, but no replies).
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Levi-Civita connection and Christoffel symbols
  1. Levi Civita symbol (Replies: 20)

Loading...