Killing fields on incomplete manifolds need not generate global isometries

On a geodesically complete Riemannian manifold, every Killing vector field is complete and integrates to a global one-parameter subgroup of its isometry group. On the incomplete interval $(0,1)$ with its Euclidean metric, ${\partial }_x$ satisfies the Killing equation, but translations leave the interval in finite time. The full isometry group is only the identity and reflection. Local solutions of the Killing equation therefore need not equal the Lie algebra of global isometries without completeness.