Talk:Symmetric space

Definition

As stated the definition reads "Let M be a connected Riemannian manifold and p a point of M. A map f defined on a neighborhood of p is said to be a geodesic symmetry, if it fixes the point p and reverses geodesics through that point." What does 'reverses geodesics' mean; if one takes a geodesic exp(tX)p and maps it to exp(-tX/2)p, is that a reversal? If so, how is it possible that all geodesic symmetries be isometric? —Preceding unsigned comment added by 81.210.239.103 (talk) 16:34, 30 July 2008 (UTC)

No, because if you reverse it a second time, you would get exp(+tX/4)p so that won't work. 67.198.37.16 (talk) 01:59, 7 November 2020 (UTC)

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)

A couple of nice definitions are given in

"Geometry of weakly symmetric spaces" by J\"urgen Berndt and Lieven Vanhecke, J. Math. Soc. Japan 48(4) 1996, pp. 745-760.

They are the Riemannian manifolds such that for any two distinct points p and q, there is an isometry sending p to q and q to p. Equivalently, for any point and any maximal geodesic through that point, there is an isometry fixing the point and inducing a nontrivial involution of the geodesic. Equivalently again, for any point and any nonzero tangent vector to that point, there is an isometry fixing the point whose differential has eigenvalue -1 on the tangent vector. The importance is apparently that these spaces are "commutative" i.e., their invariant differential operators form a commutative algebra.

I'm not sure if it is worth to include as it isn't close enough to my research area. Nilradical (talk) 13:25, 17 September 2008 (UTC)

There's now a section on this, as well as a stub weakly symmetric spaces. It would be nice to have an example of a weakly symmetric space that is not symmetric. I'm wildly guessing that weakly-symmetric spaces are not always locally symmetric. Right? If so, and example for that would be nice... 67.198.37.16 (talk) 02:10, 7 November 2020 (UTC)

Three classes

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)

Fixed. I'm amazed this goof survived for so long! It's a good opportunity to add information about the rank too. Nilradical (talk) 13:25, 17 September 2008 (UTC)

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)

Loos and axiomatic definition

Comment: This article is mathematical correct - even more than most articles in the Wikipedia. But there is a huge But: The reference to the definition of symmetric spaces by O. Loos is completely underrepresented. He defines a symmetric space axiomatically as a set with a composition with three axioms - like the notion of a a group. If this set is a (compatible) differentiable manifold, there is a local structure in tangent space - a Lie triple. Like for a Lie group there is a Lie algebra. Note that groups in his sence become symmetric spaces, spheres and hyperboloids as well. And there is a natural symmetric structure in pseudo-orthogonal vector spaces - outside the null cone. This should have physical applications. H. Tilgner —Preceding unsigned comment added by 79.214.168.28 (talk) 12:19, 9 June 2010 (UTC)

To be a little more precise: A symmetric space is a topological space with a composition subject to three algebraic axioms and one topological one (isolation needs a topology). The three algebraic ones are (i) xx=x (ii) x(xy)=y (iii) x(yz)=(xy)(xz). How every such symmetric space can be realized by a topological group is the content of Loos' books. So this is considerably more general than here in the article. Examples are given by groups (topological or even discrete ones), Jordan algebras, the latter being constructed by real pseudo-orthogonal vector spaces and even Hilbert spaces (including those of quantum mechanics) for instance. His chapter on locally symmetric spaces is genuine general relativity. 184.22.189.33 (talk) 09:06, 18 February 2018 (UTC)

Huh. So, axiom (iii) resembles the S combinator in that (S x y z) = (x z (y z)) and the second axiom resembles the K combinator (K x y) = x and axiom one resembles (I x) = x. These are of course, the axioms of lambda calculus. (Simply typed) lambda calculus is famously the internal language of Cartesian-closed category (CCC). And so this is not entirely a surprise, because, sure, Riemannian geometry smells CCC-like ... so I'm guessing this resemblance is not at all accidental, but because (I'm guessing wildly now) that the Loos definition of symmetric spaces really is a CCC!? Oh, but this can't be right, because if Loos is lumping hilbert spaces in there, that cannot work, because hilbert spaces use Linear logic as the internal language, and not lambda calc. Oh well. Still, it smells like something worth clarifying!?67.198.37.16 (talk) 02:51, 7 November 2020 (UTC)

Definition?

The section "Definition using geodesic symmetries" fails to actually define what is a "symmetric space". Is it supposed to be the same as a "(globally) Riemannian symmetric" Riemannian manifold M? --Roentgenium111 (talk) 20:36, 1 July 2011 (UTC)

It seems to define things now. 67.198.37.16 (talk) 02:54, 7 November 2020 (UTC)

Lede

This article begins as follows:

In differential geometry, representation theory and harmonic analysis,

The terms "representation theory" and "harmonic analysis" contribute nothing to informing the lay reader that mathematics is what this is about. "Representation theory" could be about political science and "harmonic analysis" could be about music. Only the word "geoemetry" does that. Michael Hardy (talk) 16:01, 29 March 2013 (UTC)

I am fixing this now. 67.198.37.16 (talk) 02:54, 7 November 2020 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Symmetric space/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Field could also be algebra (repn theory) or (harmonic) analysis, but at the moment the geometrical viewpoint dominates. Maybe this should be balanced as the article expands. Geometry guy 20:55, 17 September 2008 (UTC)

Last edited at 20:55, 17 September 2008 (UTC). Substituted at 02:37, 5 May 2016 (UTC)

reversed geodesic?

Okay, in "it fixes the point p and reverses geodesics through that point,"what does "reverses the geodesic" mean?

A geodesic is a "straight line" in the space. It is not a vector, it is a path. You can follow the path in either direction, so in what sense is the geodesic reversed?

Also, could one half of the symmetry be seen as the other half turned inside out (point inverted)? — Preceding unsigned comment added by Verdana Bold (talk • contribs) 02:56, 12 February 2017 (UTC)

The symmetry s fixing p is just a central symmetry in normal coordinates around p, i.e. if c is a geodesic through p parametrised by arclength and such that c(0) = p then s(c(t)) = c(-t) for t in an appropriate interval around 0. jraimbau (talk) 17:08, 12 February 2017 (UTC)

Confusing notation in Riemannian classification table

In the table in Symmetric space: Classification result the entry for DIII is listed as having rank [n/2]. This notation is not explained in the article, and I am having difficulty determining what it means (is it possibly either the floor or the ceiling of the division?). It should probably be made explicit in the article by someone who knows what the right answer is better than I do.

Azaghal of Belegost (talk) 21:58, 23 May 2017 (UTC)

Azaghal, for a real number x, the square bracket notation [x] means the greatest integer that does not exceed x. Nowadays, it has largely been replaced by the computer science notation $\operatorname {floor} (x)=\lfloor x\rfloor$.2600:1700:E1C0:F340:A983:7FD0:5C4E:20BD (talk) 04:08, 24 December 2018 (UTC)

Finite covers

The 2nd paragraph says "every symmetric space... has a finite cover which is simply connected..." Why finite? The very first example - the torus - has infinite fundamental group... L3erdnik (talk) 00:59, 20 March 2020 (UTC)

Tori are not symmetric, only locally symmetric. The associated symmetric space is euclidean space (which is simply connected). jraimbau (talk) 12:09, 21 March 2020 (UTC)

Earlier the symmetric spaces are defined as for each point possessing an isometry inducing reflection about that point, flat tori satisfy that definition. L3erdnik (talk) 23:02, 21 March 2020 (UTC)

What you are claiming is false : flat tori possess only finitely many isometries whose linear part is nontrivial. So at most finitely many points can be symmetric. jraimbau (talk) 12:24, 22 March 2020 (UTC)

Let's go through this. Flat torus is a quotient R^n/Z^n for some embedding of Z^n. Reflection about point x (y->2x-y) translates (by conjugation) the action of z into action of -z, ie it preserves the set by which the quotient is taken. Therefore the reflection descends to an isometry of the quotient - the torus. L3erdnik (talk) 14:51, 22 March 2020 (UTC)

Also, tori are homogeneous spaces, all their points are equivalent. If there is a reflection about some point (which there is), there is a reflection about any other point (translate, reflect, translate back). L3erdnik (talk) 14:54, 22 March 2020 (UTC)

You're right, I was confused about conjugation vs. translation for a moment. I guess the sentence you took issue with should be amended to say that it applies only to spaces without euclidean factors or something to this effect. jraimbau (talk) 12:09, 24 March 2020 (UTC)

Looks like L3erdnik fixed it back in March. The fix looks good to me. All other mentions of "cover" in the article talk about the compact case. 67.198.37.16 (talk) 05:53, 7 November 2020 (UTC)

Geometric definition of global symmetric

The definition says that a locally symmetric space is (globally) symmetric if the geodesic symmetries, which are isometries defined on a neighourbood of the point p by the locally symmetric definition, can be extended to all of M. But I think with this definition they might be not isometries on all of M which they certainly need to be in order to define a globally symmetric space. Symmetric space means the geodesic symmetries can be extended to all of M where they define (global) isometries. The condition on the global isometry is not given automatically for a geodesic symmetry (initially defined on a neighbourhood of p) which extends to all of M, or am I wrong with this? Nevertheless, I have already changed it in the article. If I am wrong, then please tell me and just rechange the definition like before. PS: Is this definition of a global symmetric space even compatible with the one of Milne on page 8 of https://www.jmilne.org/math/xnotes/svi.pdf ? He requires the isometry to be (globally) of order 2. This article does not require such a global condition on the order. Or is this automatically satisfied because the extended geodesic symmetry is locally of order 2 and with other properties ...? --77.3.205.164 (talk) 12:42, 17 February 2021 (UTC)

"Group action having no fixed points" should really be a "free group action", no?

I'm a bit concerned about talking about "groups actions with no fixed points" when we really mean free group action. Firstly, this doesn't look like standard terminology as far as I can tell. My additional concern is that the terminology isn't literally true, given that the identity element of a group has got only fixed points. Is there any area where this term is standard? Thanks. --Svennik (talk) 09:41, 24 August 2023 (UTC)