# Localization formula on Riemannian foliations

## GAP Seminar

## Meeting Details

For more information about this meeting, contact Ping Xu, Nigel Higson, Mathieu Stienon, Kendra Stauffer, Matthew Peddie.

**Speaker:** Yi Lin, Georgia Southern University

**Abstract:** A Riemannian foliation is a foliation on a smooth manifold that comes equipped with a transverse Riemannian metric: a fiberwise Riemannian metric $g$ on the normal bundle of the foliation, such that for any vector field $X$ tangent to the leaves, the Lie derivative $L(X)g=0$. In this talk, we would discuss the notion of transverse Lie algebra actions on Riemannian foliations, which is used as a model for Lie algebra actions on the leave spaces. Using an equivariant version of the basic cohomology theory on Riemannian foliations, we explain that when the action preserves the transverse Riemannian metric, there is a foliated version of the classical Borel-Atiyah-Segal localization theorem. After a review of the transverse integration theory for basic forms on Riemannian foliations, we would also explain how to establish a foliated version of the Atiyah-Bott-Berline-Vergne integration formula, which reduce the integral of an equivariant basic cohomology class to an integral over the set of invariant leaves. This talk is based on a very recent joint work with Reyer Sjamaar.

## Room Reservation Information

**Room Number:** 320 Whitmore

**Date:** 03/20/2018

**Time:** 2:30pm - 3:30pm