### Context

#### Manifolds and cobordisms

#### Differential geometry

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.

# Contents

## 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

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

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 $(G,H)$, $(G',H')$ 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, 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 $x,y,z\in M$

- $x \cdot x = x$ (idempotence)
- $x \cdot (x\cdot y) = y$
- $x\cdot (y \cdot z) = (x \cdot y)\cdot (x \cdot z)$ (left self-distributivity)
- 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 $a \triangleright - : Q \to Q$.

## 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