User:Rybu/lie group geometric constructions

''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 $$L_g(x) = gx$$ 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, $$T_eG$$. 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 $$q_x(g) = g.x$$, 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 $$Stab(p)$$.