Symmetric space

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 $M$ be a globally symmetric Riemannian space, $G$ the connected component of the group of isometries of $M$ and $H$ the isotropy subgroup of $G$ of some point of $M$. Then definitions can be given for $M$ being of compact, non-compact or Euclidean type in terms of the corresponding pair of Lie algebras $(\mathfrak{g}, \mathfrak{h})$. In particular, if $M$ is of the non-compact type, then $\mathfrak{g}$ has a Cartan decomposition $\mathfrak{g} = \mathfrak{h} + \mathfrak{m}$, 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)