# Holonomy groups of compact Riemannian symmetric spaces

Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page:

https://en.wikipedia.org/wiki/Symmetric_space

In the Berger classifiction of holonomy groups

https://en.wikipedia.org/wiki/Holonomy#The_Berger_classification

the symmetric case is omitted because

the holonomy group can easily be read off the Cartan classification in the symmetric space case.

How does this "reading off" work exactly. Can someone point to me a list of the holonomy groups of the compact symmetric spaces?

• In general, the holonomy group and the isotropy group have the same identity component (this is a theorem of E. Cartan). So if you assume that $M$ is simply-connected, they are equal.
– abx
Jul 14, 2018 at 16:03
• Plase put this as the answer and I will accept. Jul 14, 2018 at 16:07

At the request of the OP I put my comment as an answer: in general, the holonomy group and the isotropy group have the same identity component (this is a theorem of E. Cartan). So if you assume that $M$ is simply-connected, they are equal. You can see a proof (for instance) in section 10.79 of Arthur Besse's Einstein manifolds.