A Connection Whose Curvature Is the Lie Bracket

Let G be a Lie groupwith Lie algebra g. On the trivial principal G-bundle over g there is a natural connection whose curvature is the Lie bracket of g. The exponential map of G is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold M , the curvature of the natural connection is the Lie bracket of vectorfields on M . In the case that G = SO(3) the motion of a sphere rolling on a plane is given by parallel transport of a pullback of the natural connection by a map from the plane to so(3). The motion of a sphere rolling on an oriented surface in R can be described by a similar connection. 1. A NATURAL CONNECTION AND ITS CURVATURE Samelson [4] has shown that the covariant derivative of a connection can be expressed as a Lie bracket. It is the purpose of this article to show that the Lie bracket of a Lie algebra can be expressed as the curvature form of a natural connection. The setting for this result is the following. Let π : P → X be a right principal G-bundle with the Lie group G as the structure group. A connection on P is a smoothG-equivariant distribution of horizontal spaces in the tangent bundle TP complementary to the vertical tangent spaces of the fibers. The curvature of a connection is a g-valued 2-form on the total space P . A good reference is Bleecker’s book [2]. LetG be a Lie group and g its Lie algebra. LetP = g×G be the total space of the trivial right principal G-bundle with projection P → g : (x, g) 7→ x. The right action of G on P is given by (x, h) · g = (x, hg) and let Rg denote the action of g on P ; that is, Rg : P → P : (x, h) 7→ (x, hg). Let 1 ∈ G be the identity element and let ι : g → g ×G : x 7→ (x, 1) be the identity section of the bundle. Let Lg : G → G : h 7→ gh and Rg : G → G : h 7→ hg be the left and right multiplication by g. The context should make it clear whether Rg is acting Date: August 24, 2008. 2000 Mathematics Subject Classification. Primary 53C05; Secondary 53C29.