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 separation axioms.

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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 $\sigma :G\to G$, 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

$$a▹-:G/H\to G/H$$a \triangleright - : G/H \to G/H

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 $(G,H)$, $(G\prime ,H\prime )$ can give what is normally considered the same symmetric space $G/H\cong G\prime /H\prime$. In other words, not every morphism of symmetric spaces arises from a morphism of symmetric pairs.

To avoid this problem, symmetric space is (equivalent to) a smooth manifold $M$ with multiplication $\cdot :M\times M\to M$ which is a smooth map such that for all $x,y,z\in M$

1. $x\cdot x=x$ (idempotence)
2. $x\cdot (x\cdot y)=y$
3. $x\cdot (y\cdot z)=(x\cdot y)\cdot (x\cdot z)$ (left self-distributivity)
4. for every $x$ there is a neighborhood $U\subset M$ such that $x\cdot y=y$ 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 $a▹-:Q\to Q$.

References

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, Berlin, 2000.

The relation to quandles is given in Theorem I.4.3. Bertram attributes this result to part I, chapter II of

• Ottmar Loos, Symmetric Spaces I, II, Chapter II, Benjamin, New York, 1969.
• Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces,
• S. Helgason, Group representations and symmetric spaces, Proc. ICM. Nice 1970, vol. 2, 313-320, pdf, djvu