On the relation between connection and sprays

Abstract

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)).