Talk:Riemannian symmetric space

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[edit] Weakly Riemannian symmetric space

Without a definition, that section is useless, even if the reference is valid. Arthur Rubin | (talk) 00:31, 17 February 2006 (UTC)

Agree. I've removed it for now. -- Fropuff 00:41, 17 February 2006 (UTC)

When dividing (irreducible) symmetric spaces in three classes one has to replace positive by non-negative and negative by non-positive

I transcribe this text

1. Euclidean type: M has vanishing curvature, and is therefore isometric to a Euclidean space.

2. Compact type: M has positive (SHOULD BE NON-NEGATIVE) sectional curvature.

3. Non-compact type: M has negative (SHOULD BE NON-POSITIVE) sectional curvature —Preceding unsigned comment added by 190.137.12.134 (talk) 17:14, 26 February 2008 (UTC)

[edit] Riemannian symmetric spaces vs non-compact simple Lie groups

When we compare table on this page with #REDIRECT List of simple Lie groups then we see that there are exactly 12 exceptional non-compact simple Lie groups and 12 exceptional Riemannian symmetric spaces and the dimensions also match ! In the other article there is written

"The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each non-compact simple Lie group G, one compact and one non-compact. The non-compact one is a cover of the quotient of G by a maximal compact subgroup H, and the compact one is a cover of the quotient of the compact form of G by the same subgroup H. This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry."

Can we have some note about this here as well ?

I have also second question. Having simple Lie group G and it's subgroup H. When G/H is Riemmanian symmetric space (RSS) ? E.g. E6/F4 is RSS but E7/E6 is not.

Question three: In most of the cases maximal torus in the subgroup H is the same as in group G e.g. E7/E6xSO(2) but not always. The other cases are SO8/SO7, E6/F4. Is it important feature or not ? Marek Mitros

193.41.170.225 (talk) 08:39, 16 May 2008 (UTC)