User:Rybu/lie group geometric constructions

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The point of this page is to be an omnibus page for various geometric constructions on lie groups: haar measure, left, right and bi-invariant metrics, metrics on homogeneous spaces, etc. This is a general purpose page on how lie groups give rise to various geometric objects This page will ultimately be linked to from a page on Geometric Manifolds.

A left-invariant metric on a Lie Group G is a Riemann metric on G such that the left-multiplication maps L_g : G \to G given by Lg(x) = gx are isometries. Every Lie Group has a left-invariant metric, and there is a canonical bijection between the left-invariant metrics on G and the metrics on the Lie algebra associated to G, TeG. The bijection is defined by right multiplication. If m : T_eG \oplus T_eG \to \mathbb R is a metric, a Riemann metric is defined on G by defining \mu(v,w) = m(TL_{g^{-1}}v,TL_{g^{-1}}w) provided v,w \in T_gG.


  • Background (2) -- The natural metrics on a homogeneous space: Given a Lie Group G together with a transitive action of G on a smooth manifold M, provided the point-stabilizers of this action are compact, there is a naturally-defined Riemann metric on M making the maps q_x : G \to M Riemannian submersions, where qx(g) = g.x, where G has a left-invariant metric. The basic construction is that a tangent vector to a point p \in M corresponds (via the implicit function theorem) to a normal vector field to Stab(p) \subset G. Since Stab(p) is compact, the inner product of two tangent vectors at p \in M can be defined by integrating the inner products of the corresponding normal vector fields to Stab(p).