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:
“The love of truth, virtue, and the happiness of mankind are specious pretexts, but not the inward principles that set divines at work; else why should they affect to abuse human reason, to disparage natural religion, to traduce the philosophers as they universally do?”
—George Berkeley (16851753)
“When and under what conditions is the black man to have a free ballot? When is he in fact to have those full civil rights which have so long been his in law?”
—Benjamin Harrison (18331901)