nLab symmetric space

For a topological space satisfying the $R_0$ regularity condition (which states that the specialisation preorder is symmetric, hence an equivalence relation), see symmetric topological space.

Idea

A symmetric space is a specially nice homogeneous space, characterized by the property that for each point there is a symmetry fixing that point and acting as $-1$ on its tangent space. An example would be the sphere, the Euclidean plane, or the hyperbolic plane.

Definitions

A symmetric space is classically defined to be a quotient manifold of the form $G/H$, where $G$ is a Lie group and the subgroup $H$ is the set of fixed points of some involution $\sigma \colon G \longrightarrow G$, that is, a smooth homomorphism with $\sigma^2 = 1_G$. Using the involution, every point $a \in G/H$ gives rise to a smooth function

$$a \triangleright (-) \;\colon\; G/H \longrightarrow G/H \mathrlap{\,,}$$

fixing the point $a$ and acting as $-1$ on the tangent space of $a$. This operations satisfies the laws of an involutory quandle.

More precisely, a symmetric pair is a pair $(G,H)$ where $G$ is a Lie group and the subgroup $H$ is the set of fixed points of some involution $\sigma : G \to G$. Different pairs $(G,H)$, $(G',H')$ can give what is normally considered the same symmetric space $G/H \cong G'/H'$. In other words, not every morphism of symmetric spaces arises from a morphism of symmetric pairs.

To avoid this problem, we can define a symmetric space as a smooth manifold $M$ with a smooth map $\triangleright : M\times M\to M$ such that for all $x,y,z\in M$

1. $x \triangleright x = x$ (idempotence)
2. $x \triangleright (x\triangleright y) = y$
3. $x \triangleright (y \triangleright z) = (x \triangleright y)\triangleright (x \triangleright z)$ (left self-distributivity)
4. for every $x$ there is a neighborhood $U\subset M$ such that $x \triangleright y = y$ implies $x = y$ for all $z\in U$.

This amounts to an involutory quandle object $Q$ in the category of smooth manifolds, with the property that each point $a \in Q$ is an isolated fixed point of the map $a \triangleright - : Q \to Q$.

Related entries

• homogeneous space

• quandle

• Lie triple system

• transvection

References

Original discussion and classification of symmetric spaces:

• Élie Cartan: Sur une classe remarquable d’espaces de Riemann, Bulletin de la Société Mathématique de France 54 (1926) 214-264 [numdam:BSMF_1926__54__214_0, eudml:86508]

Further discussion:

• S. Araki: On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962) 1–34

• Ottmar Loos, Symmetric Spaces I: General Theory, Benjamin (1969) [pdf]

• Ottmar Loos, Symmetric Spaces II: Compact spaces and classification, Benjamin (1969)

• Sigurdur Helgason, Group representations and symmetric spaces, Proc. Internat. Congress Math. Nice 1970, Vol. 2 book no 10, Gauthier-Villars (1971) 313-320 [pdf, pdf, djvu]

• Sigurdur Helgason, Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs 39 (1994) [doi:10.1090/surv/039]

• Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]

• Wolfgang Bertram: Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, Memoirs of the American Mathematical Society 192 (2008) [doi:10.1090/memo/0900, arXiv:math/0502168]

The definition in terms of quandles coincides with the classical definition in the case of connected symmetric spaces. For details, including a comparison of other definitions of symmetric space, see:

• Wolgang Bertram, The geometry of Jordan and Lie structures, Lecture Notes in Mathematics 1754, Springer (2000) [doi:10.1007/b76884]

  The relation to quandles is given in Theorem I.4.3. where this result is attributed to chapter II of Loos 1969 I.