On the relation Between connection and sprays

A well-known (and simple) result of linear algebra asserts that any bilinear symmetric form RⁿxR^{→ δ} Rⁿ is completely determined by its values on the diagonal: letting σ (t) = δ (t,t) be the quadratic form associated to δ, we may recover the original form δ simply as the polar form of σ, i.e. δ (t₁,t₂) = ½ (σ(t₁+t₂)- σ (t₁)- σ (t₂)).