# Integration on a Compact Connected Lie Group

# Integration on a Compact Connected Lie Group

This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group *G* acts smoothly on the left on a manifold *M*. Given any *C∞* differential *k*-form ω on *M*, by averaging all the left translates of ω over *G*, one can produce a *C∞* invariant *k*-form on *M*. As another example, on a *G*-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.

*Keywords:*
compact connected Lie group, compact Lie group, bi-invariant top-degree form, averaging, averaging operator, invariant objects, Riemannian metric

Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.