# nLab generalized tangent bundle

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

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

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

## 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

${g}_{TX}:X\to B\mathrm{GL}\left(n\right)$g_{T X} : X \to \mathbf{B} GL(n)

to the smooth moduli stack of $\mathrm{GL}\left(n\right)$.

Similarly there is a canonical morphism

$\left({g}_{TX},{g}_{TX}^{-T}\right):X\to BO\left(n,n\right)$(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 $O\left(n,n\right)$. This classifies the $O\left(n,n\right)$-principal bundle to which $TX\oplus {T}^{*}X$ is associated.

### Reduction of structure group

Where a reduction of the structure group of the tangent bundle along $BO\left(n\right)↪B\mathrm{GL}\left(n\right)$ is equivalently a vielbein/orthogonal structure/Riemannian metric on $X$, so a reduction of the structure group of the generalized tangent bundle along $B\left(O\left(n\right)×O\left(n\right)\right)\to BO\left(n,n\right)$ is a generalized vielbein, defining a type II geometry on $X$.

Other reductions yield other geometric notions, for instance:

• reduction along $U\left(n,n\right)\to O\left(2n,2n\right)$ is a generalized complex structure;

• further reduction along $\mathrm{SU}\left(n,n\right)\to U\left(n,n\right)\to O\left(2n,2n\right)$ is a generalized Calabi-Yau manifold structure.

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