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Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. They were first studied extensively and classified by Élie Cartan .


BASIC DEFINITIONS


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. It necessarily acts as -\mathrm{id} on the Tangent Space of ''p''.
Note that in general, ''f'' does not need to be isometric, nor can it in general be extended to be defined on all of ''M''.

''M'' is said to be locally Riemannian symmetric if its geodesic symmetries are in fact isometric. By virtue of the local version of the Theorem Of Cartan-Ambrose-Hicks , ''M'' is locally Riemannian symmetric if and only if its curvature tensor is Covariantly Constant .

''M'' is said to be (globally) Riemannian symmetric if it is locally Riemannian symmetric and its geodesic symmetries are defined on all of ''M''. Because of the global version of the Theorem Of Cartan-Ambrose-Hicks , any Simply Connected , Complete locally Riemannian symmetric space is actually Riemannian symmetric.


EXAMPLES


The most obvious examples of Riemannian symmetric spaces are the Euclidean Space , Sphere s, Projective Space s, and Hyperbolic Space s, each with their natural Riemannian metric. More examples are provided by compact, semi-simple Lie Groups equipped with a bi-invariant Riemannian metric.

There are many locally symmetric spaces that are not symmetric spaces.
Examples are open subsets of symmetric spaces and quotients of symmetric spaces by groups of isometries with no fixed points.

Any compact Riemann surface
of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space.


ELEMENTARY FACTS


Any Riemannian symmetric space ''M'' is complete and Riemannian Homogeneous (meaning that the isometry group of ''M'' acts transitively on ''M''). In fact, already the neutral component of the isometry group acts transitively on ''M'' (because ''M'' is connected).


THE TRANSITION FROM GEOMETRY TO ALGEBRA: DESCRIPTION VIA LIE GROUPS


Besides the view of a Riemannian symmetric space taken above, namely of a Riemannian manifold with a specific property, there is another, more algebraic view of Riemannian symmetric spaces. The transition between these two views is now described:

Let ''M'' be a Riemannian symmetric space and let ''G'' denote the neutral component of the isometry group of ''M''; ''G'' can be shown to be a Lie group Automorphism such that the isotropy group ''K'' is contained between the fixed point group ''F'' of \sigma and its neutral component. Now the "algebraic data" (G,K,\sigma,g) contain all information which is needed to reconstruct the Riemannian symmetric space ''M''.

To describe the process by which this reconstruction is achieved, i.e. the reversal of the preceding construction, we suppose that any data set (G,K,\sigma,g) is given, where ''G'' is a connected, semi-simple Lie group, ''K'' is a compact Lie subgroup of ''G'', \sigma is an involutive Lie group automorphism on ''G'' such that ''K'' is contained between the fixed point group ''F'' of \sigma and its neutral component, and ''g'' is a Riemannian metric on the quotient space ''M := G/K'' (which carries a differentiable structure because of the compactness of ''K''). In this setting ''M'' is a Riemannian manifold, and for any point ''p'' in ''M'', say ''p = fK'' with some ''f'' in ''G'', the map s_p: M o M, f'K \mapsto f\cdot \sigma(f^{-1}f')\cdot K is a well-defined isometry with s_p(p)=p and d_p s_p = -\mathrm{id}_{T_pM}, hence s_p is the geodesic symmetry of ''M'' at ''p''. This shows that ''M'' indeed becomes a Riemannian symmetric space.

If one starts with a Riemannian symmetric space ''M'', and then performs these two constructions in sequence, then the Riemannian symmetric space yielded is isometric to the original one. This shows that the "algebraic data" (G,K,\sigma,g) indeed describe the structure of the Riemannian symmetric space ''M'' in full.


STRUCTURE AND CLASSIFICATION


As Élie Cartan discovered in 1913, the algebraic description of Riemannian symmetric spaces given in the preceding section permits a full classification of the Riemannian symmetric spaces.

Let a Riemannian symmetric space ''M'' be given, and (G,K,\sigma,g) the algebraic data associated to it. We want to classify the possible isometry classes of ''M''.

First, the Universal Covering manifold of a Riemannian symmetric space is again Riemannian symmetric, and the covering map is described by dividing ''G'' by a subgroup of its center. Therefore we may suppose without loss of generality that ''M'' is simply connected. This causes ''G'' to be simply connected and ''K'' to be connected.

A simply connected Riemannian symmetric space is said to be irreducible if it is not the product of two or more Riemannian symmetric spaces. It can then be shown that any simply connected Riemannian symmetric space is the product of irreducible ones. Therefore we may further restrict ourselves to classifying the irreducible, simply connected Riemannian symmetric spaces.

The next step is to show that any irreducible, simply connected Riemannian symmetric space ''M'' is of either of the following three types:

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

2. Compact type: ''M'' has positive sectional curvature.

3. Non-compact type: ''M'' has negative sectional curvature.

The spaces of Euclidean type are completely described by the fact that they are isometric to a Euclidean space. Therefore it remains to classify the irreducible, simply connected Riemannian symmetric spaces of compact type, resp. of non-compact type.

In the following, we describe the classification for the compact type; for the non-compact type, analogous statements hold. The irreducible, simply connected Riemannian symmetric spaces of compact type fall into two categories:

a. ''G'' is a simply connected, compact simple Lie group;

b. ''M'' itself is a compact, simply connected simple Lie groups, equipped with a bi-invariant Riemannian metric, in this case ''G'' is not simple, to the contrary we have G = M imes M and K is the diagonal in this product.

Of type b, there are the classical Lie groups \mathrm{SO}(n), \mathrm{SU}(n), \mathrm{Sp}(n), as well as the five Exceptional Lie Groups ''E6'', ''E7'', ''E8'', ''F4'', ''G2''. (For the details of this classification, see Simple Lie Group .)

In the case of type a ''G'' is a simply connected, compact, simple Lie group, and therefore one of the groups cited in the preceding paragraph. Moreover, ''K'' equals the neutral component of the fixed point group of the involution \sigma. Therefore it suffices to classify the involutive automorphisms of each of the compact simple Lie groups (up to conjugation); each such automorphism will give rise to one Riemannian symmetric space of compact type.

In this way, Cartan found that there are the following seven infinite series and twelve exceptional Riemannian symmetric spaces of type a. They are here given as quotient spaces ''G/K'', together with a geometric interpretation, if readily available. The names of these spaces were assigned by Cartan.


SYMMETRIC SPACES AND HOLONOMY


Suppose that the connected component of the Holonomy Group of a Riemannian manifold at a point acts irreducibly on the tangent space. Then either the manifold is a locally symmetric space, or it is in one of 7 Families .


HERMITIAN SYMMETRIC SPACES


A Riemannian symmetric space which is additionally equipped with a parallel complex structure compatible with the Riemannian metric is called Hermitian symmetric space. Some examples are complex vector spaces and complex projective spaces, both with their usual Riemannian metric, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric.

From the classification of the Riemannian symmetric spaces, it can be readily read off which of the spaces are in fact Hermitian symmetric. In both the compact and the non-compact cases it turns out that there are four infinite series, namely AIII, BDI with ''p=2'', DIII and CII, and two exceptional spaces, namely EIII and EVII.
The non-compact Hermitian symmetric spaces can be realized as Bounded Domain s in complex vector spaces.


EXTENSIONS OF THE CONCEPT


The concept of a Riemannian symmetric space can be generalized by replacing the Riemannian metric with a Pseudo-Riemannian Metric (i.e. lifting the requirement that the metric be positive definite), thereby obtaining pseudo-Riemannian symmetric spaces. An example is the 1-sheeted hyperboloid in flat Lorentz space.

Lorentzian symmetric spaces, i.e. pseudo-Riemannian symmetric spaces of signature 1,
occasionally appear in General Relativity (though most interesting spacetimes are not symmetric spaces).

It should be noted that the classification of pseudo-Riemannian symmetric spaces is much more complicated than that of Riemannian symmetric spaces, and only specific cases have been treated by now. The main reason for this is that the isometry group of a pseudo-Riemannian symmetric space is not necessarily semi-simple, and therefore one is unable to base a classification on the known classification of semi-simple Lie groups.

An even more general concept is that of an affine symmetric space. Here the manifold ''M'' is only equipped with a Covariant Derivative ; we call ''M'' affine symmetric, if its geodesic symmetries are all globally defined and affine maps.


FURTHER READING


  • Kobayashi, Nomizu, ''Foundations of Differential Geometry, Volume II''. ISBN 0-471-15732-5. Chapter XI contains a good introduction to Riemannian symmetric spaces.

  • Helgason, ''Differential geometry, Lie groups, and symmetric spaces''. ISBN 0-8218-2848-7. The standard book on Riemannian symmetric spaces.

  • Loos, ''Symmetric spaces I: General Theory''.

  • Loos, ''Symmetric spaces II: Compact Spaces and Classification''.

  • Besse, ''Einstein manifolds'' ISBN 0-387-15279-2.