Localization formulas about two Killing vector fields

In this article, we will discuss the smooth (XM+ √ −1YM )-invariant forms on M and to establish a localization formulas. As an application, we get a localization formulas for characteristic numbers. The localization theorem for equivariant differential forms was obtained by Berline and Vergne(see [2]). They discuss on the zero points of a Killing vector field. Now, We will discuss on the points about two Killing vector fields and to establish a localization formulas. Let M be a smooth closed oriented manifold. Let G be a compact Lie group acting smoothly on M , and let g be its Lie algebra. Let g be a G-invariant metric on TM . If X, Y ∈ g, let XM , YM be the corresponding smooth vector field on M . If X, Y ∈ g, then XM , YM are Killing vector field. Here we will introduce the equlvariant cohomology by two Killing vector fields. 1 Equlvariant cohomology by two Killing vector fields First, let us review the definition of equlvariant cohomology by a Killing vector field. Let Ω∗(M) be the space of smooth differetial forms on M , the de Rham complex is (Ω∗(M), d). Let LXM be the Lie derivative of XM on Ω ∗(M), iXM be the interior multiplication induced by the contraction of XM . Set dX = d+ iXM , then dX = LXM by the following Cartan formula LXM = [d, iXM ]. Let ΩX(M) = {ω ∈ Ω∗(M) : LXMω = 0} be the space of smooth XM -invariant forms on M . Then d 2 Xω = 0, when ω ∈ ΩX(M). It is a complex (ΩX(M), dX). The corresponding cohomology group H∗ X(M) = KerdX|Ω∗X(M) ImdX|Ω∗X(M) is called the equivariant cohomology associated with X. If a form ω has dXω = 0, then ω called dX-closed form. ∗Email: [email protected]