Why do those two terms add here?

[Emilie.Jung]

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?

 

[andrewkirk]

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.

 

[Emilie.Jung]

I maybe should then move this post to another section @andrewkirk

 

[PeterDonis]

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