nLab
exterior bundle

Contents

Definition

For XX a smooth manifold its exterior bundle is the vector bundle T *X\wedge^\bullet T^\ast X which is the direct sum (Whitney sum) of the bundle of differential n-forms for all nn \in \mathbb{N}.

Properties

If (X,g)(X,g) is a Riemannian manifold, then the exterior bundle supports a canonical Dirac operator, namely the Kähler-Dirac operator d+d d + d^\dagger. The corresponding Fredholm operator (d+d *)(1+(d+d *) 2) 1/2(d+ d^\ast)(1 + (d + d^\ast)^2)^{-1/2} constitutes a canonical class in the K-homology of XX.

Created on May 20, 2013 at 11:48:42. See the history of this page for a list of all contributions to it.