# Riemannian symmetric space of dimension $n$ and rank $n-1$

I need to cover a case of $$n$$-dimensional locally symmetric Riemannian space of rank $$n-1$$. Is there a simple proof that there is no such irreducible space ($$n>4$$)? If I need to cite the Cartan classification for that, which concrete paper/book you suggest?

• Multiply hyperbolic plane (or 2d sphere) by the Euclidean space of suitable dimension. Sep 17, 2020 at 3:51
• @MoisheKohan yes, but what about irreducible ones? Sep 17, 2020 at 13:05
• Just go through the list at en.wikipedia.org/wiki/Symmetric_space and references therein. Sep 17, 2020 at 18:48

An irreducible symmetric space $$M$$ of dimension $$n$$ and rank $$n-1$$ has dimension $$n=2$$. Indeed, the rank is the codimension of a principal orbit of the isotropy group $$K$$ at $$p \in M$$ acting on the tangent space $$T_pM$$. So the $$K$$-orbits are one dimensional. Now any monoparametric subgroup of $$K$$ has a one dimensional orbit contained in a $$2$$-dimensional subspace of $$T_pM$$ hence $$K$$ has a one dimensional orbit contained in a $$2$$-dimensional subspace of $$T_pM$$. So $$K$$ has an invariant $$2$$-dimensional subspace in $$T_pM$$. But $$K$$ is the holonomy group of $$M$$ at $$p$$ hence $$K$$ acts irreducibly on $$T_pM$$. Then $$dim(T_p M)=2$$.