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For now, I have problem with the beginning of the section 2:

The metric of the near-extremal D3-brane is

## ds_{10}^2=H^{-\frac 1 2} (-h dt^2+d\vec x ^2)+H^{\frac 1 2} (\frac{dr^2}{h}+d\Omega_5^2) \ \ \ \ \ \ \ \left( H=1+(\frac{L}{r})^4 \ \ , \ \ h=1-(\frac{r_H}{r})^4 \right) \ \ \ \ \ (1)##

Here ##\vec x = (x, y, z) ## are the spatial coordinates along which the D3-brane is extended and ##dΩ^2_5## is the standard metric on the five-sphere ##S^5## with unit radius. The near-horizon limit consists of “dropping the 1” from H. Then the metric is ##AdS_5-Schwarzschild##,

##ds_5^2=(\frac{r}{L})^2 (-hdt^2+d\vec x^2)+(\frac{L}{r})^2 \frac{dr^2}{h} \ \ \ \ \ \ \ \ \ \ (2)##

times the metric for an ##S^5## of constant radius L.

I don't understand how he got equation (2) from equation (1) by just "dropping the 1 from H"! Because when I do that, I get:

## (\frac r L)^2 ( -h dt^2+ d \vec x ^2)+(\frac L r)^2 ( \frac{dr^2}{h}+d\Omega_5^2) ##

And this is really a mystery to me how that ## d\Omega_5^2 ## vanishes and reappears as an overall multiplicative factor! This really strange because in equation (1), the "radius" of the ##S^5## is ## H^{\frac 1 2} ## but in the near-horizon limit, it involves other coordinates too, actually their differentials which is non-sense because terms like ## dt^2 d\Omega_5^2 ## will appear in the metric. What is the author doing here?

Thanks