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Abstract: We study the characteristics of affine vector fields on pseudo-Riemannian manifolds and provide tensorial formulas that characterize these vector fields. Using our approach, we present a straightforward proof that any affine vector field on a compact Riemannian manifold is a Killing vector field. Moreover, we establish several necessary and sufficient conditions for an affine vector field on a Riemannian manifold to be classified as a Killing or parallel vector field.