6
$\begingroup$

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?

Improve this question
$\endgroup$
2
  • 3
    $\begingroup$ 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. $\endgroup$
    – abx
    Commented Jul 14, 2018 at 16:03
  • $\begingroup$ Plase put this as the answer and I will accept. $\endgroup$
    – Pierre Dubois
    Commented Jul 14, 2018 at 16:07

1 Answer 1

Reset to default
6
$\begingroup$

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.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .