Affine Differential Geometry - Two Natural Conditions

Two Natural Conditions

We impose two natural conditions. The first is that the induced connexion ∇ and the induced volume form ω be compatible, i.e. ∇ω ≡ 0. This means that ∇_Xω = 0 for all X ∈ Ψ(M). In other words, if we parallel transport the vectors X₁,…,X_n along some curve in M, with respect to the connexion ∇, then the volume spanned by X₁,…,X_n, with respect to the volume form ω, does not change. A direct calculation shows that ∇_Xω = τ(X)ω and so ∇_Xω = 0 for all X ∈ Ψ(M) if, and only if, τ ≡ 0, i.e. D_Xξ ∈ Ψ(M) for all X ∈ Ψ(M). This means that the derivative of ξ, in a tangent direction X, with respect to D always yields a, possibly zero, tangent vector to M. The second condition is that the two volume forms ω and ν coincide, i.e. ω ≡ ν.