The rank of a symmetric space

This question reminds me of when I was a graduate student. At some point Gelfand asked me "What is the rank of a symmetric space" and I just spat back the usual definition, something like what Matrix found in Wikipedia. Gelfand shook his head as if I had said something really stupid and proceeded to explain:

Euclidean space, hyperbolic space, complex projective space (and so on) are rank one. Why? Because if you have two pairs of points and the distance between them is the same, then there is an isometry that takes one pair of points to the other. ONE invariant is all you need to determine whether two pairs of points are the same up to isometries.

The Grassmannian of two-planes in ${\mathbb R}^4$ has rank two : you need two invariants to determine if two pairs of points are equivalent up to isometry. Take two planes in four-space passing through the origin. Draw a circle with center zero in one plane. Project it orthogonally onto the second plane. You get an ellipse, but you cannot compare it to the circle because it lives on a different plane so project it back to the first plane. The minor and major axes of your ellipse (with respect to the circle) are two invariants that are preserved by any isometry of the pair of planes. Conversely if you have two pairs of planes that have the same two invariants, then there is an isometry of the Grassmannian that takes one pair of planes to the other.

I went back home and the uninsightful book I was reading on symmetric spaces went back to the library the next day.

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).

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.