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,…,Xn along some curve in M, with respect to the connexion ∇, then the volume spanned by X1,…,Xn, 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. DXξ ∈ 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.
Read more about this topic: Affine Differential Geometry
Famous quotes containing the words natural and/or conditions:
“Men nowhere, east or west, live yet a natural life, round which the vine clings, and which the elm willingly shadows. Man would desecrate it by his touch, and so the beauty of the world remains veiled to him. He needs not only to be spiritualized, but naturalized, on the soil of earth.”
—Henry David Thoreau (18171862)
“My dear young friend ... civilization has absolutely no need of nobility or heroism. These things are symptoms of political inefficiency. In a properly organized society like ours, nobody has any opportunities for being noble or heroic. Conditions have got to be thoroughly unstable before the occasion can arise.”
—Aldous Huxley (18941963)