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A general name given to various types of spaces in differential geometry.
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A general name given to various types of spaces in differential geometry.<ol><li style="margin-bottom: 10px;">A manifold with an affine connection is called a locally symmetric affine space if the [[Torsion tensor|torsion tensor]] and the covariant derivative of the [[Curvature tensor|curvature tensor]] vanish identically.</li><li style="margin-bottom: 10px;">A (pseudo-) [[Riemannian manifold|Riemannian manifold]] is called a locally symmetric (pseudo-) Riemannian space if the covariant derivative of its curvature tensor with respect to the Levi-Civita connection vanishes identically.</li><li style="margin-bottom: 10px;"> A pseudo-Riemannian manifold (respectively, a manifold with an affine connection) $M$ is called a globally symmetric pseudo-Riemannian (affine) space if one can assign to every point $x \in M$ an isometry (affine transformation) $S_x$ of $M$ such that $S_x^2 = id$ and $x$ is an isolated fixed point of $S_x$.</li><li style="margin-bottom: 10px;"> Let $G$ be a connected Lie group, let $\Phi$ be an involutive automorphism (i.e. $\Phi^2 = id$), let $G^\Phi$ be the closed subgroup of all $\Phi$-fixed points, let $G_0^\Phi$ be the component of the identity in $G^\Phi$, and let $H$ be a closed subgroup of $G$ such that$$G_0^\Phi \subset H \subset G^\Phi$$Then the homogeneous space $G/H$ is called a symmetric homogeneous space.</li><li style="margin-bottom: 10px;"> A symmetric space in the sense of Loos (a Loos symmetric space) is a manifold $M$ endowed with a multiplication$$M \times M \longrightarrow M, \qquad (x,y) \mapsto x.y$$satisfying the following four conditions:<ol style="list-style-type: lower-alpha;"><li> $x.x=x$;</li><li>$x.(x.y)=y$;</li><li>$x.(y.z)=(x.y).(x.z)$;</li><li>every point $x \in M$ has a neighbourhood $U$ such that $x.y=y$ implies $y=x$ for all $y \in U$.</li></ol>Any globally symmetric affine (pseudo-Riemannian) space is a locally symmetric affine (pseudo-Riemannian) space and a homogeneous symmetric space. Any homogeneous symmetric space is a globally symmetric affine space and a Loos symmetric space. Every connected Loos symmetric space is a homogeneous symmetric space.Let $M$ be a connected Loos symmetric space, and hence a homogeneous space: $M=G/H$. Then $G/H$ can be equipped with a torsion-free invariant affine connection with the following properties:<ol style="list-style-type: lower-roman;">  <li>the covariant derivative of the curvature tensor vanishes;</li>      <li>every geodesic $\gamma$ is a trajectory of some one-parameter subgroup $\psi$ of $G$, and parallel translation of vectors along $\gamma$ coincides with their translation by means of $\psi$; and</li>      <li>the geodesics are closed under multiplication (they are called one-dimensional subspaces).</li></ol>Similarly one can introduce the concept of an arbitrary subspace of $M$, namely, a manifold $N$ of $M$ which is closed under multiplication and which is a symmetric space under the induced multiplication. A closed subset $N$ of $M$ which is stable under multiplication is a subspace. </br></br>The analogue of the Lie algebra for a symmetric space $G/H$ is defined as follows: Let $\mathfrak{g}$ and $\mathfrak{h}$ be the Lie algebras of the groups  $G$ and $H$, respectively, and let $\phi = d\Phi_e$ (the differential at the unit), where  $\Phi$ is the involutive automorphism defining the symmetric homogeneous space $G/H$. The eigenvectors of the space endomorphism $\phi$ corresponding to the eigenvalue $-1$ form a subspace $\mathfrak{m}$ such that $\mathfrak{g}$ is the direct sum of the subspaces$\mathfrak{m}$  and $\mathfrak{h}$, and  can be identified with the tangent space of $G/H$ at the point $0=H$. If one defines a trilinear composition on the vector space $\mathfrak{m}$ by$$\mathfrak{m} \times \mathfrak{m} \times \mathfrak{m}  \longrightarrow \mathfrak{m}, \qquad \left(X,Y,Z \right) \mapsto R \left(X,Y \right) Z,$$where $R$ is the curvature tensot, then $ \mathfrak{m}$ becomes a [[Lie ternary system|Lie ternary system]].  If $N$ is a subspace of $M$ passing through the point $0$, then the tangent space of $N$ at $0$ is a subsystem of $ \mathfrak{m}$ and conversely.If $M$ is a Loos symmetric space, then so is the product $M \times M$. Let $R$ be a subspace of $M \times M$ defining an equivalence relation on $M$. Then $R$ is called a congruence. This concept is used in the construction of a theory of coverings for symmetric spaces. Two points $x,y \in M$ are said to commute if$$x.(a.(y.b)) = y.(a.(x.b)) \qquad \text{for all}\; a,b \in M.$$The centre $Z(M)$ of $M$ with respect to a point $0 \in M$ is defined to be the set of all points of $M$ which commute with $0$. $Z(M)$ is a closed subspace of $M$ which can be equipped with an Abelian group structure. Let $M$ be a simply-connected symmetric space. Then the search for symmetric spaces for which $M$ is a covering space reduces to the classification of discrete subgroups of $Z(M)$.In the theory of symmetric spaces, considerable attention is devoted to classification problems (see ). Let $M$ be a locally symmetric Riemannian space. Then $M$ is called reducible if, in some coordinate system, its fundamental quadratic form can be written as$$ds^2 = g_{ij}\left(x^1, \dots, x^k  \right) dx^i dx^j+g_{\alpha\beta} \left(x^{k+1}, \dots, x^n  \right) dx^\alpha dx^\beta,$$$$i,j = 1, \dots, k ; \qquad \alpha, \beta = k+1 , \dots , n.$$Otherwise the space is called irreducible. E. Cartan has shown that the study of all irreducible locally symmetric Riemannian spaces reduces to the classification of involutive automorphisms of real compact Lie algebras, which he accomplished. At the same time he solved the local classification problem for symmetric homogeneous spaces whose fundamental groups are simple and compact. A classification of symmetric homogeneous spaces with simple non-compact fundamental groups has been obtained (see  [[#References|[3]]], [[#References|[5]]]).</li></ol>
  
1) A manifold with an affine connection is called a locally symmetric affine space if the [[Torsion tensor|torsion tensor]] and the covariant derivative of the [[Curvature tensor|curvature tensor]] vanish identically.
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====References====
 
 
2) A (pseudo-) [[Riemannian manifold|Riemannian manifold]] is called a locally symmetric (pseudo-) Riemannian space if the covariant derivative of its curvature tensor with respect to the Levi-Civita connection vanishes identically.
 
 
 
3) A pseudo-Riemannian manifold (respectively, a manifold with an affine connection) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s0917101.png" /> is called a globally symmetric pseudo-Riemannian (affine) space if one can assign to every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s0917102.png" /> an isometry (affine transformation) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s0917103.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s0917104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s0917105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s0917106.png" /> is an isolated fixed point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s0917107.png" />.
 
 
 
4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s0917108.png" /> be a connected Lie group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s0917109.png" /> be an involutive automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171010.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171011.png" /> be the closed subgroup of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171012.png" />-fixed points, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171013.png" /> be the component of the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171014.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171015.png" /> be a closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171016.png" /> such that
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171017.png" /></td> </tr></table>
 
 
 
Then the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171018.png" /> is called a symmetric homogeneous space.
 
 
 
5) A symmetric space in the sense of Loos (a Loos symmetric space) is a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171019.png" /> endowed with a multiplication
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171020.png" /></td> </tr></table>
 
 
 
satisfying the following four conditions:
 
 
 
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171021.png" />;
 
 
 
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171022.png" />;
 
 
 
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171023.png" />;
 
 
 
d) every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171024.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171026.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171028.png" />.
 
 
 
Any globally symmetric affine (pseudo-Riemannian) space is a locally symmetric affine (pseudo-Riemannian) space and a homogeneous symmetric space. Any homogeneous symmetric space is a globally symmetric affine space and a Loos symmetric space. Every connected Loos symmetric space is a homogeneous symmetric space.
 
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171029.png" /> be a connected Loos symmetric space, and hence a homogeneous space: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171030.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171031.png" /> can be equipped with a torsion-free invariant affine connection with the following properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171032.png" />) the covariant derivative of the curvature tensor vanishes; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171033.png" />) every geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171034.png" /> is a trajectory of some one-parameter subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171036.png" />, and parallel translation of vectors along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171037.png" /> coincides with their translation by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171038.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171039.png" />) the geodesics are closed under multiplication (they are called one-dimensional subspaces). Similarly one can introduce the concept of an arbitrary subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171040.png" />, namely, a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171042.png" /> which is closed under multiplication and which is a symmetric space under the induced multiplication. A closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171044.png" /> which is stable under multiplication is a subspace. The analogue of the Lie algebra for a symmetric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171045.png" /> is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171047.png" /> be the Lie algebras of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171049.png" />, respectively, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171050.png" /> (the differential at the unit), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171051.png" /> is the involutive automorphism defining the symmetric homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171052.png" />. The eigenvectors of the space endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171053.png" /> corresponding to the eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171054.png" /> form a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171056.png" /> is the direct sum of the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171058.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171059.png" /> can be identified with the tangent space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171060.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171061.png" />. If one defines a trilinear composition on the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171062.png" /> by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171063.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171064.png" /> is the curvature tensor, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171065.png" /> becomes a [[Lie ternary system|Lie ternary system]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171066.png" /> is a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171067.png" /> passing through the point 0, then the tangent space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171068.png" /> at 0 is a subsystem of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171069.png" /> and conversely.
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171070.png" /> is a Loos symmetric space, then so is the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171071.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171072.png" /> be a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171073.png" /> defining an equivalence relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171074.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171075.png" /> is called a congruence. This concept is used in the construction of a theory of coverings for symmetric spaces. Two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171076.png" /> are said to commute if
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171077.png" /></td> </tr></table>
 
 
 
The centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171078.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171079.png" /> with respect to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171080.png" /> is defined to be the set of all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171081.png" /> which commute with 0. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171082.png" /> is a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171083.png" /> which can be equipped with an Abelian group structure. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171084.png" /> be a simply-connected symmetric space. Then the search for symmetric spaces for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171085.png" /> is a covering space reduces to the classification of discrete subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171086.png" />.
 
 
 
In the theory of symmetric spaces, considerable attention is devoted to classification problems (see ). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171087.png" /> be a locally symmetric Riemannian space. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171088.png" /> is called reducible if, in some coordinate system, its fundamental quadratic form can be written as
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171089.png" /></td> </tr></table>
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171090.png" /></td> </tr></table>
 
 
Otherwise the space is called irreducible. E. Cartan has shown that the study of all irreducible locally symmetric Riemannian spaces reduces to the classification of involutive automorphisms of real compact Lie algebras, which he accomplished. At the same time he solved the local classification problem for symmetric homogeneous spaces whose fundamental groups are simple and compact. A classification of symmetric homogeneous spaces with simple non-compact fundamental groups has been obtained (see , [[#References|[3]]], [[#References|[5]]]).
 
 
====References====
 
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,  "Selected works on geometry" , Kazan'  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  E. Cartan,  "Sur une classe rémarkable d'espaces de Riemann"  ''Bull. Soc. Math. France'' , '''54'''  (1926)  pp. 214–264</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  E. Cartan,  "Sur une classe rémarkable d'espaces de Riemann"  ''Bull. Soc. Math. France'' , '''55'''  (1927)  pp. 114–134</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Berger,  "Les espaces symmétriques noncompacts"  ''Ann. Sci. École Norm. Sup.'' , '''74'''  (1957)  pp. 85–177</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  O. Loos,  "Symmetric spaces" , '''1–2''' , Benjamin  (1969)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,  "Selected works on geometry" , Kazan'  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  E. Cartan,  "Sur une classe rémarkable d'espaces de Riemann"  ''Bull. Soc. Math. France'' , '''54'''  (1926)  pp. 214–264</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  E. Cartan,  "Sur une classe rémarkable d'espaces de Riemann"  ''Bull. Soc. Math. France'' , '''55'''  (1927)  pp. 114–134</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Berger,  "Les espaces symmétriques noncompacts"  ''Ann. Sci. École Norm. Sup.'' , '''74'''  (1957)  pp. 85–177</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  O. Loos,  "Symmetric spaces" , '''1–2''' , Benjamin  (1969)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
  
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====Comments====
  
 
====Comments====
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171091.png" /> be a globally symmetric Riemannian space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171092.png" /> the connected component of the group of isometries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171094.png" /> the isotropy subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171095.png" /> of some point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171096.png" />. Then definitions can be given for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171097.png" /> being of compact, non-compact or Euclidean type in terms of the corresponding pair of Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171098.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171099.png" /> is of the non-compact type, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s091710100.png" /> has a [[Cartan decomposition|Cartan decomposition]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s091710101.png" />, see [[#References|[5]]].
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171091.png" /> be a globally symmetric Riemannian space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171092.png" /> the connected component of the group of isometries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171094.png" /> the isotropy subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171095.png" /> of some point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171096.png" />. Then definitions can be given for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171097.png" /> being of compact, non-compact or Euclidean type in terms of the corresponding pair of Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171098.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s09171099.png" /> is of the non-compact type, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s091710100.png" /> has a [[Cartan decomposition|Cartan decomposition]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091710/s091710101.png" />, see [[#References|[5]]].
 +
====References====
  
====References====
 
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.L. Besse,  "Einstein manifolds" , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.L. Besse,  "Einstein manifolds" , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>

Revision as of 22:13, 15 October 2012

A general name given to various types of spaces in differential geometry.

  1. A manifold with an affine connection is called a locally symmetric affine space if the torsion tensor and the covariant derivative of the curvature tensor vanish identically.
  2. A (pseudo-) Riemannian manifold is called a locally symmetric (pseudo-) Riemannian space if the covariant derivative of its curvature tensor with respect to the Levi-Civita connection vanishes identically.
  3. A pseudo-Riemannian manifold (respectively, a manifold with an affine connection) $M$ is called a globally symmetric pseudo-Riemannian (affine) space if one can assign to every point $x \in M$ an isometry (affine transformation) $S_x$ of $M$ such that $S_x^2 = id$ and $x$ is an isolated fixed point of $S_x$.
  4. Let $G$ be a connected Lie group, let $\Phi$ be an involutive automorphism (i.e. $\Phi^2 = id$), let $G^\Phi$ be the closed subgroup of all $\Phi$-fixed points, let $G_0^\Phi$ be the component of the identity in $G^\Phi$, and let $H$ be a closed subgroup of $G$ such that$$G_0^\Phi \subset H \subset G^\Phi$$Then the homogeneous space $G/H$ is called a symmetric homogeneous space.
  5. A symmetric space in the sense of Loos (a Loos symmetric space) is a manifold $M$ endowed with a multiplication$$M \times M \longrightarrow M, \qquad (x,y) \mapsto x.y$$satisfying the following four conditions:
    1. $x.x=x$;
    2. $x.(x.y)=y$;
    3. $x.(y.z)=(x.y).(x.z)$;
    4. every point $x \in M$ has a neighbourhood $U$ such that $x.y=y$ implies $y=x$ for all $y \in U$.
    Any globally symmetric affine (pseudo-Riemannian) space is a locally symmetric affine (pseudo-Riemannian) space and a homogeneous symmetric space. Any homogeneous symmetric space is a globally symmetric affine space and a Loos symmetric space. Every connected Loos symmetric space is a homogeneous symmetric space.Let $M$ be a connected Loos symmetric space, and hence a homogeneous space: $M=G/H$. Then $G/H$ can be equipped with a torsion-free invariant affine connection with the following properties:
    1. the covariant derivative of the curvature tensor vanishes;
    2. every geodesic $\gamma$ is a trajectory of some one-parameter subgroup $\psi$ of $G$, and parallel translation of vectors along $\gamma$ coincides with their translation by means of $\psi$; and
    3. the geodesics are closed under multiplication (they are called one-dimensional subspaces).
    Similarly one can introduce the concept of an arbitrary subspace of $M$, namely, a manifold $N$ of $M$ which is closed under multiplication and which is a symmetric space under the induced multiplication. A closed subset $N$ of $M$ which is stable under multiplication is a subspace.

    The analogue of the Lie algebra for a symmetric space $G/H$ is defined as follows: Let $\mathfrak{g}$ and $\mathfrak{h}$ be the Lie algebras of the groups $G$ and $H$, respectively, and let $\phi = d\Phi_e$ (the differential at the unit), where $\Phi$ is the involutive automorphism defining the symmetric homogeneous space $G/H$. The eigenvectors of the space endomorphism $\phi$ corresponding to the eigenvalue $-1$ form a subspace $\mathfrak{m}$ such that $\mathfrak{g}$ is the direct sum of the subspaces$\mathfrak{m}$ and $\mathfrak{h}$, and can be identified with the tangent space of $G/H$ at the point $0=H$. If one defines a trilinear composition on the vector space $\mathfrak{m}$ by$$\mathfrak{m} \times \mathfrak{m} \times \mathfrak{m} \longrightarrow \mathfrak{m}, \qquad \left(X,Y,Z \right) \mapsto R \left(X,Y \right) Z,$$where $R$ is the curvature tensot, then $ \mathfrak{m}$ becomes a Lie ternary system. If $N$ is a subspace of $M$ passing through the point $0$, then the tangent space of $N$ at $0$ is a subsystem of $ \mathfrak{m}$ and conversely.If $M$ is a Loos symmetric space, then so is the product $M \times M$. Let $R$ be a subspace of $M \times M$ defining an equivalence relation on $M$. Then $R$ is called a congruence. This concept is used in the construction of a theory of coverings for symmetric spaces. Two points $x,y \in M$ are said to commute if$$x.(a.(y.b)) = y.(a.(x.b)) \qquad \text{for all}\; a,b \in M.$$The centre $Z(M)$ of $M$ with respect to a point $0 \in M$ is defined to be the set of all points of $M$ which commute with $0$. $Z(M)$ is a closed subspace of $M$ which can be equipped with an Abelian group structure. Let $M$ be a simply-connected symmetric space. Then the search for symmetric spaces for which $M$ is a covering space reduces to the classification of discrete subgroups of $Z(M)$.In the theory of symmetric spaces, considerable attention is devoted to classification problems (see ). Let $M$ be a locally symmetric Riemannian space. Then $M$ is called reducible if, in some coordinate system, its fundamental quadratic form can be written as$$ds^2 = g_{ij}\left(x^1, \dots, x^k \right) dx^i dx^j+g_{\alpha\beta} \left(x^{k+1}, \dots, x^n \right) dx^\alpha dx^\beta,$$$$i,j = 1, \dots, k ; \qquad \alpha, \beta = k+1 , \dots , n.$$Otherwise the space is called irreducible. E. Cartan has shown that the study of all irreducible locally symmetric Riemannian spaces reduces to the classification of involutive automorphisms of real compact Lie algebras, which he accomplished. At the same time he solved the local classification problem for symmetric homogeneous spaces whose fundamental groups are simple and compact. A classification of symmetric homogeneous spaces with simple non-compact fundamental groups has been obtained (see [3], [5]).

References

[1] P.A. Shirokov, "Selected works on geometry" , Kazan' (1966) (In Russian)
[2a] E. Cartan, "Sur une classe rémarkable d'espaces de Riemann" Bull. Soc. Math. France , 54 (1926) pp. 214–264
[2b] E. Cartan, "Sur une classe rémarkable d'espaces de Riemann" Bull. Soc. Math. France , 55 (1927) pp. 114–134
[3] M. Berger, "Les espaces symmétriques noncompacts" Ann. Sci. École Norm. Sup. , 74 (1957) pp. 85–177
[4] O. Loos, "Symmetric spaces" , 1–2 , Benjamin (1969)
[5] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)

Comments

Let be a globally symmetric Riemannian space, the connected component of the group of isometries of and the isotropy subgroup of of some point of . Then definitions can be given for being of compact, non-compact or Euclidean type in terms of the corresponding pair of Lie algebras . In particular, if is of the non-compact type, then has a Cartan decomposition , see [5].

References

[a1] A.L. Besse, "Einstein manifolds" , Springer (1987)
[a2] B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
How to Cite This Entry:
Symmetric space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_space&oldid=28422
This article was adapted from an original article by A.S. Fedenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article