nLab symmetric space



Manifolds and cobordisms

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


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 0R_0 regularity condition (which states that the specialisation preorder is symmetric, hence an equivalence relation), see symmetric topological space.



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 1-1 on its tangent space. An example would be the sphere, the Euclidean plane, or the hyperbolic plane.


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

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

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

More precisely, a symmetric pair is a pair (G,H)(G,H) where GG is a Lie group and the subgroup HH is the set of fixed points of some involution σ:GG\sigma : G \to G. Different pairs (G,H)(G,H), (G,H)(G',H') can give what is normally considered the same symmetric space G/HG/HG/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 MM with a smooth map :M×MM\triangleright : M\times M\to M such that for all x,y,zMx,y,z\in M

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

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


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.

See also:

  • 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

Last revised on May 23, 2023 at 22:18:46. See the history of this page for a list of all contributions to it.