special holonomy


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




tangent cohesion

differential cohesion

graded differential 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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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)

Differential cohomology



For XX a space equipped with a GG-connection on a bundle \nabla (for some Lie group GG) and for xXx \in X any point, the parallel transport of \nabla assigns to each curve Γ:S 1X\Gamma : S^1 \to X in XX starting and ending at xx an element hol (γ)G hol_\nabla(\gamma) \in G: the holonomy of \nabla along that curve.

The holonomy group of \nabla at xx is the subgroup of GG on these elements.

If \nabla is the Levi-Civita connection on a Riemannian manifold and the holonomy group is a proper subgroup HH of the special orthogonal group, one says that (X,g)(X,g) is a manifold of special holonomy .



Berger's theorem says that if a manifold XX is

then the possible special holonomy groups are the following

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
\mathbb{C}Kähler manifoldU(k)2k2kKähler forms ω 2\omega_2
Calabi-Yau manifoldSU(k)2k2k
\mathbb{H}quaternionic Kähler manifoldSp(k)Sp(1)4k4kω 4=ω 1ω 1+ω 2ω 2+ω 3ω 3\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3
hyper-Kähler manifoldSp(k)4k4kω=aω 2 (1)+bω 2 (2)+cω 2 (3)\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2 (a 2+b 2+c 2=1a^2 + b^2 + c^2 = 1)
𝕆\mathbb{O}Spin(7) manifoldSpin(7)8Cayley form
G2 manifoldG277associative 3-form

Relation to GG-reductions

A manifold having special holonomy means that there is a corresponding reduction of structure groups.


Let (X,g)(X,g) be a connected Riemannian manifold of dimension nn with holonomy group Hol(g)O(n)Hol(g) \subset O(n).

For GO(n)G \subset O(n) some other subgroup, (X,g)(X,g) admits a torsion-free G-structure precisely if Hol(g)Hol(g) is conjugate to a subgroup of GG.

Moreover, the space of such GG-structures is the coset G/LG/L, where LL is the group of elements suchthat conjugating Hol(g)Hol(g) with them lands in GG.

This appears as (Joyce prop. 3.1.8)

Via 𝕆\mathbb{O}-Riemannian manifolds

normed division algebra𝔸\mathbb{A}Riemannian 𝔸\mathbb{A}-manifoldsSpecial Riemannian 𝔸\mathbb{A}-manifolds
real numbers\mathbb{R}Riemannian manifoldoriented Riemannian manifold
complex numbers\mathbb{C}Kähler manifoldCalabi-Yau manifold
quaternions\mathbb{H}quaternion-Kähler manifoldhyperkähler manifold

(Leung 02)


The classification in Berger's theorem is due to

  • M. Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955)

For more see

  • Nigel Hitchin, Special holonomy and beyond, Clay Mathematics Proceedings (pdf)

  • Dominic Joyce, Compact manifolds with special holonomy , Oxford Mathematical Monographs (2000)

  • Luis J. Boya, Special Holonomy Manifolds in Physics Monografías de la Real Academia de Ciencias de Zaragoza. 29: 37–47, (2006). (pdf)

Discussion of the relation to Killing spinors includes

Discussion in terms of Riemannian geometry modeled on normed division algebras is in

See also

  • Hans Freudenthal, Lie groups in the foundations of geometry, Advances in Mathematics, volume 1, (1965) pp. 145 - 190 (dspace)

Revised on July 16, 2016 13:29:03 by Anonymous Coward (