# Symmetric space

From Encyclopedia of Mathematics

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

Symmetric space.

- 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.
- 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.
- 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 $S_x$.
- 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.
- 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:
- $x.x=x$;
- $x.(x.y)=y$;
- $x.(y.z)=(x.y).(x.z)$;
- every point $x \in M$ has a neighbourhood $U$ such that $x.y=y$ implies $y=x$ for all $y \in U$.

- the covariant derivative of the curvature tensor vanishes;
- 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
- the geodesics are closed under multiplication (they are called one-dimensional subspaces).

#### 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) |

**How to Cite This Entry:**

Symmetric space.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Symmetric_space&oldid=31224

This article was adapted from an original article by A.S. Fedenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article