# 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, $T X$ its tangent bundle and $T^* X$ the cotangent bundle that the fiberwise direct sum-bundle $T X \oplus T^* X$ is the generalized tangent bundle.

More generally, a vector bundle $E \to X$ that sits in an exact sequence $T^* X \to E \to T X$ 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

$g_{T X} : X \to \mathbf{B} GL(n)$

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

Similarly there is a canonical morphism

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

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

### Reduction of structure group

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

Other reductions yield other geometric notions, for instance:

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