nLab
generalized tangent bundle

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

In the context of generalized complex geometry one says for XX a manifold, TXT X its tangent bundle and T *XT^* X the cotangent bundle that the fiberwise direct sum-bundle TXT *XT X \oplus T^* X is the generalized tangent bundle.

More generally, a vector bundle EXE \to X that sits in an exact sequence T *XETXT^* X \to E \to T X is called a generalized tangent bundle, such as notably those underlying a Courant Lie 2-algebroid over XX.

Properties

As an associated bundle

The ordinary tangent bundle is the canonical associated bundle to the general linear group-principal bundle classified by the morphism

g TX:XBGL(n) g_{T X} : X \to \mathbf{B} GL(n)

to the smooth moduli stack of GL(n)GL(n).

Similarly there is a canonical morphism

(g TX,g TX T):XBO(n,n) (g_{T X}, g^{-T}_{T X}) : X \to \mathbf{B} O(n,n)

to the moduli stack which is the delooping of the Narain group O(n,n)O(n,n). This classifies the O(n,n)O(n,n)-principal bundle to which TXT *XT X \oplus T^* X is associated.

Reduction of structure group

Where a reduction of the structure group of the tangent bundle along BO(n)BGL(n)\mathbf{B} O(n) \hookrightarrow \mathbf{B} GL(n) is equivalently a vielbein/orthogonal structure/Riemannian metric on XX, so a reduction of the structure group of the generalized tangent bundle along B(O(n)×O(n))BO(n,n)\mathbf{B} (O(n) \times O(n)) \to \mathbf{B}O(n,n) is a generalized vielbein, defining a type II geometry on XX.

Other reductions yield other geometric notions, for instance:

Revised on May 29, 2012 06:56:57 by Urs Schreiber (131.130.239.199)