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:
“A woman with cut hair is a filthy spectacle, and much like a monster ... it being natural and comely to women to nourish their hair, which even God and nature have given them for a covering, a token of subjection, and a natural badge to distinguish them from men.”
—William Prynne (1600–1669)
“I can’t say that the college-bred woman is the most contented woman. The broader her mind the more she understands the unequal conditions between men and women, the more she chafes under a government that tolerates it.”
—Susan B. Anthony (1820–1906)