synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
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The magic algebraic facts
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differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
For a space equipped with a -connection on a bundle (for some Lie group ) and for any point, the parallel transport of assigns to each curve in starting and ending at an element : the holonomy of along that curve.
The holonomy group of at is the subgroup of on these elements.
If is the Levi-Civita connection on a Riemannian manifold and the holonomy group is a proper subgroup of the special orthogonal group, one says that is a manifold of special holonomy .
Berger's theorem says that if a manifold 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:
A manifold having special holonomy means that there is a corresponding reduction of structure groups.
Let be a connected Riemannian manifold of dimension with holonomy group .
For some other subgroup, admits a torsion-free G-structure precisely if is conjugate to a subgroup of .
Moreover, the space of such -structures is the coset , where is the group of elements suchthat conjugating with them lands in .
This appears as (Joyce prop. 3.1.8)
(Leung 02)
special holonomy, reduction of structure groups, G-structure, exceptional geometry, Walker coordinates
The classification in Berger's theorem is due to
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
On special holonomy orbifolds:
Discussion of special holonomy manifolds in supergravity and superstring theory as fiber-spaces for KK-compactifications preserving some number of supersymmetries:
Last revised on July 15, 2020 at 19:33:18. See the history of this page for a list of all contributions to it.