So we obtain D plus 1 dimensional Euclidean space and

D-dimensional sphere embedded into this space.

So what we obtain is just space of,

obviously, space of constant curvature of the radius 1 over H,

which is apparent from this equation, and

which does solve this equation in Euclidean signature.

So we can define Einstein's theory in the same

manner in space of Euclidian symmetry.

Nowhere in the duration of this equation we have used

the fact that this metric is of Minkowskian signature.

So the same procedures are applicable in Euclidian signature,

we can obtain the same equation.

And the sphere will solve this equation with plus sign here.

And during this change, weak rotation, the curvature of the spacetime is not changed.

That's the reason de Sitter space is called

the spacetime of positive constant curvature.

And that is a way to see that it is, in fact, the space of constant curvature.

That's the first way of seeing that.

The second way of seeing this is as follows.

One can see that isometry of this space

contains the following group, SO

D minus D minus 1 comma 1.

Which is nothing but the Lorentzian boosts and

rotations of this spacetime.

This group doesn't change this equation.

So this group of Lorentzian rotations of this spacetime is the isometry

group of the de Sitter spacetime, which is apparent from the equation.

At the same time, an arbitrary point,

say, for example, the point X0,

X1, and so forth, XD, well, X2,

let me write X2 also, for obvious reasons, etc.

For this point, which is 0,

1 over H0 and etc, 0, everywhere,

this point belongs to this hyperboloid.

It is moved under this group, but

remains unchanged under the action of the SO,

I should say that, well, actually,

I'm sorry, to say that this is,

the isometry group is actually SO

D1 without [INAUDIBLE] and the subgroup

of this which doesn't change this point

is actually SO D-1 S0 D-1 comma 1.

So because here D X's and 1 was minus sign X0,

that's the reason SO D is isometry of this.

And the stabilizer, so-called stabilizer, the subgroup of this group which

doesn't move arbitrary point of this spacetime, say, this one, is of this form.

Then one can see that this space that we

are discussing is homogeneous space SO(D,

1) over SO(D-1, 1).

Compare, actually, this with SO(D+1) over SO(D) for the sphere.

You see under this weak rotation,

we change this to this and this to this.

So in fact, all of these homogenous spaces transform into this.