Bi-invariant metric - Imported Knowlpedia Content

Let $G$ be a Lie group with left translations $L_g$ and right translations $R_g$ .

Definition. A Riemannian metric $\langle\cdot,\cdot\rangle$ on $G$ is bi-invariant if

L_g^*\langle\cdot,\cdot\rangle = \langle\cdot,\cdot\rangle \quad\text{and}\quad R_g^*\langle\cdot,\cdot\rangle = \langle\cdot,\cdot\rangle \qquad \text{for all } g\in G.

Lie algebra formulation. A left-invariant metric is determined by an inner product $\langle\cdot,\cdot\rangle_e$ on $\mathfrak{g}=\mathrm{Lie}(G)$ . For connected $G$ , this metric is bi-invariant if and only if $\langle\cdot,\cdot\rangle_e$ is invariant under the adjoint representation:

\langle \mathrm{Ad}_g X,\mathrm{Ad}_g Y\rangle_e = \langle X,Y\rangle_e \quad\text{for all } g\in G,\; X,Y\in \mathfrak{g}.

Remarks

Consequences. For a bi-invariant metric, geodesics through the identity are precisely one-parameter subgroups $t\mapsto \exp(tX)$ (compare the exponential/one-parameter subgroup lemma). Existence is special: every compact Lie group admits one (see bi-invariant metrics on compact Lie groups), while many non-compact groups do not.

Example of a canonical source. On a semisimple Lie algebra, the Killing form provides an $\mathrm{Ad}$ -invariant bilinear form; for compact semisimple groups, its negative is positive definite and yields a bi-invariant metric.