synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
For $X$ a space equipped with a $G$-connection on a bundle $\nabla$ (for some Lie group $G$) and for $x \in X$ any point, the parallel transport of $\nabla$ assigns to each curve $\Gamma : S^1 \to X$ in $X$ starting and ending at $x$ an element $hol_\nabla(\gamma) \in G$: the holonomy of $\nabla$ along that curve.
The holonomy group of $\nabla$ at $x$ is the subgroup of $G$ on these elements.
If $\nabla$ is the Levi-Civita connection on a Riemannian manifold and the holonomy group is a proper subgroup $H$ of the special orthogonal group, one says that $(X,g)$ is a manifold of special holonomy .
Berger's theorem says that if a manifold $X$ is
neither locally a product nor a symmetric space
then the possible special holonomy groups are the following
classification of special holonomy manifolds by Berger's theorem:
G-structure | special holonomy | dimension | preserved differential form | |
---|---|---|---|---|
$\mathbb{C}$ | Kähler manifold | U(k) | $2k$ | Kähler forms $\omega_2$ |
Calabi-Yau manifold | SU(k) | $2k$ | ||
$\mathbb{H}$ | quaternionic Kähler manifold | Sp(k)Sp(1) | $4k$ | $\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$ |
hyper-Kähler manifold | Sp(k) | $4k$ | $\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2$ ($a^2 + b^2 + c^2 = 1$) | |
$\mathbb{O}$ | Spin(7) manifold | Spin(7) | 8 | Cayley form |
G2 manifold | G2 | $7$ | associative 3-form |
A manifold having special holonomy means that there is a corresponding reduction of structure groups.
Let $(X,g)$ be a connected Riemannian manifold of dimension $n$ with holonomy group $Hol(g) \subset O(n)$.
For $G \subset O(n)$ some other subgroup, $(X,g)$ admits a torsion-free G-structure precisely if $Hol(g)$ is conjugate to a subgroup of $G$.
Moreover, the space of such $G$-structures is the coset $G/L$, where $L$ is the group of elements suchthat conjugating $Hol(g)$ with them lands in $G$.
This appears as (Joyce prop. 3.1.8)
normed division algebra | $\mathbb{A}$ | Riemannian $\mathbb{A}$-manifolds | Special Riemannian $\mathbb{A}$-manifolds |
---|---|---|---|
real numbers | $\mathbb{R}$ | Riemannian manifold | oriented Riemannian manifold |
complex numbers | $\mathbb{C}$ | Kähler manifold | Calabi-Yau manifold |
quaternions | $\mathbb{H}$ | quaternion-Kähler manifold | hyperkähler manifold |
octonions | $\mathbb{O}$ | Spin(7)-manifold | G2-manifold |
(Leung 02)
special holonomy, reduction of structure groups, G-structure, exceptional geometry, Walker coordinates
The classification in Berger's theorem is due to
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