And as a result, if we use these coordinates and

plug them into the metric of ambient space,

we get the induced metric in anti-de Sitter space,

which has the following form, 1 over H squared hyperbolic

cosine squared of rho d tau squared minus d rho squared

minus sine squared of rho d omega d minus 2 squared,

where this is the coordinates of the sphere given by this n's standing here.

This is product.

So one can see immediately that, well,

one comment which is important to say here is the following,

that if we are dealing with D = 3, so we consider

two-dimensional anti-de Sitter space, no,

if D = 2, sorry, two-dimensional anti-de Sitter space,

because then here we have D+1 of course coordinates.

For D=2, for the case of two-dimensional anti-de Sitter space,

the equation that we obtained here is very similar to the equation for

two-dimensional de-Sitter space.

In fact, it just coincides with it.

And what is important here that this factor is absent, for the case of D = 2,

we don't have this factor.

So the metric is just this.

And in fact, in this case,

one can easily see the two-dimensional anti-de Sitter space,

two-dimensional anti-de Sitter space is the same kind of hyperboloid,

but rotated by the angle p over 2, so it lays here.

So this is X0, X1, and X2.

That what happens, so the same kind of hyperboloid,

but embedded into, yeah, better to say that actually,

this is x2, and x1, because, and X,

in this coordinates, and in X0 and X2 we have a circle.

And that actually already is seen here.

This tau, which is time, is ranging from 0 to 2 pi.

So we have a compact time.

So we have closed timelike curves in anti-de Sitter space,

which is considered as physically unfavorable.

And that's the reason one usually considers universal color of this space.

So the same metric like this, but

with tau ranging from minus infinity to plus infinity.

That is what usually called by anti-de Sitter space in physics literature,

frequently called.

So, universal, how, the same metric like this, but

with time ranging not in this range, but here.

And one more comment, that in two-dimensional anti-de Sitter space,

rho here is ranging in the limits from minus infinity to plus infinity.

And constant tau slice is just hyperbola,

one of the hyperbolas lying on this hyperboloid.

And two-dimensional anti-de Sitter space has two boundaries, here and here.

While, if we consider D > 2, starting with 3 and

so forth, rho, with tau, the range of values

of tau is not dependent on the dimensionality.

It's always appropriate to have it like this, but we consider universal cover.

But what happens with rho, that's only for D=2 like this.

For D>2, rho is ranging from 0 to infinity.

And that's how we cover uniquely only once the anti-de Sitter space.

And actually, time slice, this part of the metric,

for this case has the following form.

It's 1 over H squared d rho squared plus hyperbolic

sine of rho squared d omega D-2 squared.