Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-07-06T06:17:10.684Z Has data issue: false hasContentIssue false

Differential forms and connections adapted to a contact structure, after M. Rumin

Published online by Cambridge University Press:  16 October 2009

Pierre Pansu
Affiliation:
U.R.A. 169 du C.N.R.S., Centre de Mathématiques, Ecole Polytechnique F-91128 Palaiseau, U.R.A. 1169 du C.N.R.S. Mathématiques Université Paris-Sud F-91405 Orsay
Dietmar Salamon
Affiliation:
University of Warwick
Get access

Summary

Michel Rumin is a student of Mikhael Gromov, who asked him the following question : Let M be a manifold with contact structure ξ, E a vector bundle over M. A partial connection on E is a covariant derivative ∇ve defined for smooth sections e of E but only for vectors v in ξ. In particular, parallel translation is defined only along Legendrian curves, that is curves which are tangent to ξ. Can one define the curvature of such a connection?

Gromov provided the following hint : For an ordinary connection A, curvature arises in the asymptotics of holonomy around short loops. A loop encompasses a certain “span” (a 2-vector, see below), quadratic in length, and holonomy deviates from the identity by an amount proportional to curvature times span, that is, quadratic in length. In case M has dimension 3 and carries a contact structure, then every Legendrian loop has essentially zero area. Gromov conjectured that, in this case, curvature should arise as the cubic term in the asymptotic expansion of holonomy.

Michel Rumin has found a notion of curvature for partially defined connections along the above lines. The point is to understand the exterior differential for a partially defined 1-form. In fact, M. Rumin constructs a substitute for the de Rham complex : a locally exact complex of hypoelliptic operators naturally attached to a contact manifold (M, ξ) of dimension 2m+ 1. The operator which sends m-forms to m+1-forms is new. It is of second order.

Type
Chapter
Information
Symplectic Geometry , pp. 183 - 196
Publisher: Cambridge University Press
Print publication year: 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×