I would like to work a theorem on a article who deals with the rank one symmetric spaces.


i looked up the definition of symmetric spaces of rank one, but I did not find a satisfactory definition then what is the meaning of rank, intuitively and mathematically? please if anybody already worked with rank one symmetric spaces..



The rank of a symmetric space
First the algebraic definition. A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie Group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$. 


Then the rank of a symmetric space is the dimension of the "maximal $\mathbb{R}$-split torus", i.e. the maximal dimension of an abelian diagonalizable over $\mathbb{R}$ subgroup of $G$. 


The geometric meaning is that the rank is the dimension of the maximal flat submanifold of the symmetric space. If the rank is $1$, then the maximal flats are geodesics, and the symmetric space turns out to be negatively curved. 


If the rank is larger then one, then the symmetric space is only non-positively curved. However, higher-rank symmetric spaces have spectacular rigidity properties (e.g. Margulis superrigidity, arithmeticity and the normal subgroup property come to mind). 


There are only three families of rank 1 symmetric spaces, 


1) hyperbolic $n$-space, corresponding to the Lie group $SO(n,1)$.


2) complex hyperbolic $n$-space, corresponding to the Lie group $SU(n,1)$.


3) quaternionic hyperbolic $n$-space, corresponding to the Lie group $Sp(n,1)$. 


There is also one exceptional example:


4) the Cayley upper half plane, corresponding to the Lie group $F\_4^{-20}$. 


The spaces 3) and 4) have some but not all of the rigidity properties of higher rank (in particular, superrigidity and arithmeticity, but not the normal subgroup property). 


