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An affine vector field (sometimes affine collineation or affine) is a projective vector field preserving geodesics and preserving the affine parameter. Mathematically, this is expressed by the following condition:

${\displaystyle ({\mathcal {L}}_{X}g_{ab})_{;c}=0}$

• Conformal vector field
• Curvature collineation
• Homothetic vector field
• Killing vector field
• Matter collineation
• Spacetime symmetries

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