The rest of the lecture is dedicated to the anti de sitter space time which is a solution of the Einstein equations with a minus sign. D-1,I remind you that it solves this equation. (D-1)(D-2) over H^2 G nu. All this is a metric tangent, this is a Einstein [INAUDIBLE]. So it's always a minus sign. So it's a space with constant negative curvature. And it can be obtained also geometrically as the following hyper-surface, hyperbolic, actually, also. Embedded in a bit different space time so like it is defined like this D-1 Xj squared- XD squared = -1 over H squared. Compared to the sitter space, there, if we take this to minus, to this side, this will be with plus sign. All of this will be with minus sign. And this will be again with the plus sign. We have a bit different signature here. We have two minuses and extra pluses, while in de Sitter space we had one minus and the rest was plus. So this hyperboloid is embedded into the following very strange spacetime with the following metric. Unusual so far we never encountered it. D- 1, dx, j squared, plus dx, D squared. So, it has a signature. Plus, minus, minus, minus, minus, so forth, minus, and then the final plus. So, it has two pluses, so, to say, two times, and the rest. As spatial coordinates. So let's see what happens with the space under vik rotation. So let's change XD here and here. XD to iXD. What happens, then we get here a plus sign and here a minus sign. So what is this? Then this is a regular mean called skin space. And what is this? This is actually the falling space. So after this we obtain the falling, the falling actually hyperboloid. j from 1 to D, Xj squared. Now it's runs from 1 to D. While here it was to D- 1 but was extra plus here after this rotation, we obtained this. And this is equal to 1/H squared. So what is this, what do we have? As the result, we have that sum over J. XJ-squared equal to X0-squared minus 1 over h-squared. We have already encountered this such a space in the previous lectures for the case when it was three dimensional. Or four-dimensional. Or no, three-dimensional, when D was actually ranging to 4 here. To three, sorry, to three here. In fact, this is a two-sheeted hyperboloid, because it has one sheet for x0, starting with 1/H, and this shift starting with x0 less than -1/H. We encountered it for the case when H was equal to 1. So this is a Lobachevsky space, d plus one dimensional. So we have embedded into d plus one dimensional Minkowski space. We have embedded D dimensional Lobachevsky space. D dimensional Lobachevsky space. And Lobachevsky space is known to have a constant negative curvature. And that's the reason this space time, which is called anti de Sitter space time, has constant negative curvature. So while de Sitter space was mapped under weak rotation to the sphere, on the weak rotation. And in this case we obtain hyperboloid. So actually under this kind of maps, this or similar for different coordinates and map of H to iH, we can define maps between D dimensional sphere. D dimensional Lobachevsky space. D dimensional divisor space. And D dimensional antidivisor space. All of them can be mapped by various variations of this relation. Note that this map changes the sign of carriager. So, under this map, we get plus here, instead of minus. So, all of these maps allow us to map all these spaces to each other, and actually, under this map, all the embedding spaces are mapped to each other. And also the equations are mapped to each other. So, these kind of maps allow us to understand the geometry of all these spaces in a unique manner. So we can understand geodesics as we did that for the sphere and for the citrus space. We can understand also the hyperbolic distance and the de sitter space, etc, etc. So this is a simple method to understand the geometry and physics in the spaces. Although these kind of maps do not go too far. The geometry is map but some of the physics not always is mapped on this space. Well just to list one complication on which happen which goes beyond the discussion of our lectures is that, for example, if you want to see the green functions in these spaces, there are not mapped under these maps. Well, some of them do, but not all of them. If you just naively apply this map for the green function, you obtain wrongly behaving green functions. In some cases.