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Let $M$ be a Riemannian symmetric space. $M$ is then of the form $G/H$ for a semi-simple Lie group $G$ and a closed Lie subgroup $H \subset G$. In this classification, the metric $\eta$ on $M$ is given by $$\eta_{[g]}(X,Y)=K(dl_{g^{-1}}X,dl_{g^{-1}}),$$ where $K$ is the Killing form on $\mathfrak{g}$, $g \in G$ and $X$, $Y \in T_gG$. This is well-defined (independent of choice of $g \in [g] \in G/H$), because of the $ad$-invariance of $K$.

The Riemannian symmetric spaces have been classified, there is a list of all possibilities.

Question 1: What are the holonomy groups for all symmetric spaces?

It should be not a very difficult task to compute them in all cases, but I have not found it anywhere.

Question 2: Even if $G/H$ is not a symmetric space, $G/H$ still carries the above described canonical metric, as long as $G$ is semi-simple. What are the holonomy groups in these cases?

user520548
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  • In the second to final line of this section of the holonomy article on Wikipedia, they say that the local holonomy is just $H$: https://en.wikipedia.org/wiki/Holonomy#The_Berger_classification – Ben Sheller Jan 11 '18 at 13:27
  • Fantastic, thank you, that answers the question about symmetric spaces and connected Lie groups. Even though I didn't include the word "connected" in my question, that's such a small restriction to make, that it doesn't bother me. But if I understand correctly, this does not answer the question for general homogenous spaces, right?

    And I also just noticed, that your answer is contained as a side note in the answer for the following question: https://math.stackexchange.com/questions/1918994/holonomy-of-lie-groups?rq=1 . (Not to discredit you in any way, thanks for taking the time to reply!)

     – user520548 Jan 11 '18 at 14:35

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