AdS$_4$ and $\mathbb{H}^4$: What is the difference between them?

Question

This figure (source) shows the embedding of 4D hyperbolic space $\mathbb{H}^4$ and 4D de Sitter space dS$_4$ in 5D Minkowski space $\mathbb{M}^5$. $\mathbb{H}^4$ is a hyperboloid of two sheets and dS$_4$ is a hyperboloid of one sheet. However, I also understand that 4D anti-de Sitter space AdS$_4$ can be embedded in $\mathbb{M}_5$, and that it is also hyperbolic but simply connected everywhere. I want to know why the author calls the figure on the right de Sitter space but he calls the figure on the left hyperbolic space rather than anti-de Sitter space.

Is AdS$_4$ just one of the $\mathbb{H}^4$ hyperboloids? Do the two possible hyperbolic embeddings correspond to the $\{\mp\pm\pm\pm\}$ metric signature freedom? Does $\mathbb{H}^4$ have a Lorentzian signature or is it just the Euclidean version of AdS$_4$? If so, how can I reconcile the simply connected property of AdS$_4$ with the disconnected property of $\mathbb{H}^4$?

Answer 1

$\text{AdS}_n$ is a sphere of timelike radius in a space of two timelike and $n-1$ spacelike dimensions.* $\text{AdS}_n$ itself has one timelike dimension.

For comparison:

$\mathbb H^n$ is a sphere of timelike radius in a space of one timelike and $n$ spacelike dimensions,** and has zero timelike dimensions itself;

$\text{dS}_n$ is a sphere of spacelike radius in a space of one timelike and $n$ spacelike dimensions, and has one timelike dimension itself.

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* Actually, it's usually taken to be the universal cover of that sphere, since otherwise it's periodic in time, i.e., has closed causal loops.

** Usually with opposite points identified, so that there's only one sheet.