TITLE: The rank of a symmetric space
QUESTION [21 upvotes]: 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..

REPLY [3 votes]: If $M=G/K$ has noncompact type and ${\rm rank}(M)=r\ge 1$, then the space of invariant differential operators on $M$ is generated by a $r$-tuple. One can choose the first element of this $r$-tuple to be the Laplace/Casimir operator, so for rank $1$ symmetric spaces it is always generated by the Laplacian.