# AdS space as an example of klein geometry

+ 2 like - 0 dislike
79 views

If we start with the definition of the coset $AdS_{D+2}:=\frac{O(2,D)}{O(1,D)}$ , How do we derive the constraint equation for the AdS coordinates $\mu \nu -(X^{i})^{2}=R^{2}$ ?

+ 2 like - 0 dislike

Consider a vector in $\mathbb{R}^{2,D}$ with norm $-R^{2}$ , The set of all vectors with this norm are rotated into each other by the $O(2,D)$ Rotations. Use this group to make the vector in the form $X=(1,0,0...)$ It is obvious that the isotropy group that leaves this invariant is $O(1,D)$ and thus we get the equivalence because these vectors with the specified norm are in one to one correspondance with the group transformations modulo the isotropy group.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.