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
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
In the context of generalized complex geometry one says for a manifold, its tangent bundle and the cotangent bundle that the fiberwise direct sum-bundle is the generalized tangent bundle.
More generally, a vector bundle that sits in an exact sequence is called a generalized tangent bundle, such as notably those underlying a Courant Lie 2-algebroid over .
The ordinary tangent bundle is the canonical associated bundle to the general linear group-principal bundle classified by the morphism
to the smooth moduli stack of .
Similarly there is a canonical morphism
to the moduli stack which is the delooping of the Narain group . This classifies the -principal bundle to which is associated.
Where a reduction of the structure group of the tangent bundle along is equivalently a vielbein/orthogonal structure/Riemannian metric on , so a reduction of the structure group of the generalized tangent bundle along is a generalized vielbein, defining a type II geometry on .
Other reductions yield other geometric notions, for instance:
reduction along is a generalized complex structure;
further reduction along is a generalized Calabi-Yau manifold structure.
Spin(8)-subgroups and reductions to exceptional geometry
see also: coset space structure on n-spheres
Last revised on March 30, 2019 at 14:01:06. See the history of this page for a list of all contributions to it.