Projective vector fields

A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold (p.ex. spacetime) ${\displaystyle M}$ whose flow preserves the geodesic structure of ${\displaystyle M}$ without necessarily preserving the affine parameter of any geodesic. More intuitively, the flow of the projective maps geodesics smoothly into geodesics without preserving the affine parameter.

In dealing with a vector field ${\displaystyle X}$ on a semi Riemannian manifold (p.ex. in general relativity), it is often useful to decompose the covariant derivative into its symmetric and skew-symmetric parts:

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