In general relativity, a vector field ##\xi## in a space-time ##(M,g)## that preserves the metric tensor ##g## is exactly one which satisfies ##\mathcal{L}_{\xi}g = 0## where ##\mathcal{L}_{\xi}## is the Lie derivative along ##\xi##; such a vector field is called a killing field.
As an example, take ##M = \mathbb{R}^{4}## and ##g = \eta## where ##\eta## is the Minkowski metric i.e. we are in Minkowski space-time. Then the vector fields solving ##\mathcal{L}_{\xi}\eta = 0## are given by ##\xi = AX + T##. It can be shown that ##T## is the generators of translations in Minkowski space-time and that ##A## is the generator of Lorentz transformations (i.e. Lorentz boosts and spatial rotations).
Further more, in general relativity a freely falling particle satisfies the equations of motion ##\nabla_u u = 0## where ##\nabla## is the Levi-Civita connection associated with the metric tensor ##g## of the space-time, and ##u## is the 4-velocity of the particle. If ##\xi## is a killing field then ##\nabla_u (u \cdot \xi) = \xi \cdot \nabla_u u + u \cdot \nabla_u \xi = 0## so ##u \cdot \xi## is a conserved quantity along the freely falling particle's worldline. For example if ##\xi## generates time translations then ##u \cdot \xi## represents the conserved energy of the freely falling particle.
