generalized tangent bundle
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 .
As an associated bundle
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.
Reduction of structure group
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.
Revised on May 29, 2012 06:56:57
by Urs Schreiber