Symmetry of a pseudo-Riemannian manifold

(Symmetry of a Riemannian manifold)
Definition. Let $(M, O, A, g)$ be a Riemannian manifold, and let ${X_{1}, \dots, X_{s}} \subset X (M)$ , and denote $L := {span}_{R} {X_{1}, \dots, X_{s}}$ . If $(L, [\cdot, \cdot])$ is a Lie algebra, it is said to be a symmetry of the metric manifold, if for all $X \in L$

g ((h_{λ}^{X})_{*} (A), (h_{λ}^{X})_{*} (B)) = g (A, B),

for $A, B \in T_{p} M$ and $(h_{λ}^{X})_{*}$ is the push-forward of the flow of $X$ . We can write this alternatively as

(h_{λ}^{X})^{*} g = g,

where $(h_{λ}^{X})^{*}$ is the pull-back associated to the flow of $X$ .
Equivalently,

L_{X} g = 0.

See also isometry and Killing vector field.