Left-invariant metric

A left-invariant Riemannian metric is preserved by every left translation. Its Killing fields generated by left translations are right-invariant vector fields.

A Riemannian metric on a Lie group is bi-invariant if both left and right translations are isometries. Its identity inner product is invariant under the Adjoint representation of a Lie group; infinitesimally,This skew-adjointness is the key additional property beyond left invariance.

If a Lie group has an Adjoint representation of a Lie group element with a real eigenvector of eigenvalue whose modulus differs from one, it has no bi-invariant Riemannian metric. Conjugation is a composition of left and right translations, hence would preserve the positive identity inner product. Applying it to that eigenvector gives $\Vert v\Vert =|\lambda |\Vert v\Vert$, a contradiction. The orientation-preserving affine group of the real line supplies a connected example: conjugation by dilation of factor two multiplies the infinitesimal translation by two.

For a bi-invariant Riemannian metric on a Lie group, the induced inner product on its Lie algebra makes every adjoint operator skew-adjoint. The Levi-Civita connection of a bi-invariant metric gives $R(X,Y)Z=-\frac{1}{4}[[X,Y],Z]$. Thus $K(X,Y)=\frac{1}{4}\Vert [X,Y]{\Vert }^2$ for orthonormal $X,Y$, andThe nullspace of this quadratic form is exactly the center of a Lie algebra. A zero center therefore gives positive Ricci curvature, uniformly bounded below by a positive constant times the metric through left invariance and compactness of the unit sphere in the Lie algebra.

Integral curves through the identity of left-invariant vector fields are one-parameter subgroups, by flow uniqueness and left translation. They extend for all time because a fixed local existence interval translates to every point. Under a bi-invariant Riemannian metric they are geodesics by the Levi-Civita connection of a bi-invariant metric. Geodesic uniqueness proves that these are all geodesics through the identity.

For left-invariant vector fields, the Koszul formula and the adjoint invariance of a bi-invariant Riemannian metric give the displayed Levi-Civita connection. In particular ${\nabla }_{X}X=0$.

For a left-invariant metric on a Lie group, the one-parameter subgroup ${\gamma }_{\xi }(t)=\exp (t\xi )$ is a geodesic exactly whenfor every $\eta$ in the Lie algebra. The Koszul formula gives $⟨{\nabla }_{\xi }\xi ,\eta ⟩=-⟨\xi ,[\xi ,\eta ]⟩$.