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I Why do those two terms add here?

  1. Nov 30, 2015 #1
    In complex manifold, one writes

    $$ds^2=g_{ab}dz^adz^b\hspace{3cm}(1)$$
    $$ds^2=2g_{\alpha\bar{\beta}}dz^{\alpha}d\bar{z}^{\bar{\beta}}+g_{\alpha\beta}dz^{\alpha\beta}dz^{\alpha}dz^{\beta}+g_{\bar{\alpha}\bar{\beta}}d\bar{z}^{\bar{\alpha}}d\bar{z}^{\bar{\beta}}\hspace{3cm}(2)$$

    My question is why is there a "2" in the first term? Looks like two terms added up there but how come?
     
  2. jcsd
  3. Nov 30, 2015 #2

    andrewkirk

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    When it comes to metrics, I only know about real manifolds, but I would guess that the first term is a simplification of

    $$
    g_{\alpha\bar{\beta}}dz^{\alpha}d\bar{z}^{\bar{\beta}}+
    g_{\bar{\alpha}\beta}d\bar{z}^{\bar{\alpha}}dz^\beta
    $$

    I don't know if that's equivalent to what's written, as I don't know the notation for complex metrics.
     
  4. Dec 1, 2015 #3
    I maybe should then move this post to another section @andrewkirk
     
  5. Dec 2, 2015 #4

    PeterDonis

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    Thread closed, new thread will be reposted in another subforum.
     
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