# nLab symmetric space

Contents

### Context

#### Manifolds and cobordisms

Definitions

Genera and invariants

Classification

Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

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.

# 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, 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 $a \triangleright - : Q \to Q$.

## References

• 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]

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.

Last revised on July 11, 2024 at 11:02:13. See the history of this page for a list of all contributions to it.