Metric of Lorentzian signature on a compact homogeneous space.

Question

Can anyone give an example of a metric of Lorentzian signature on a compact homogeneous space.

Answer 1

Take any compact Lie group and choose a Lorentz inner product on its Lie algebra. This determines a left-invariant (and thus homogeneous) Lorentz metric on the group.

For example, on $SU(2) \approx \mathbb S^3$, let $Z$ be the vector field that generates the Hopf action, $t \centerdot x = e^{it} x$; and let $X,Y$ be left-invariant vector fields such that $\{X,Y,Z\}$ form an orthonormal frame with respect to the round metric. Let $\{\xi,\eta,\zeta\}$ be the coframe dual to this frame. Then $g = \xi^2 + \eta^2 - \zeta^2$ is a left-invariant Lorentz metric on the group. (This is sometimes called a Lorentzian Berger sphere.)