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:
“Your child should feel entitled to cry when you leave; crying is a natural thing for a child to do when she feels bad. The fact that your child cries when you go doesn’t mean she will never like day care. It just means she wants you to stay.”
—Amy Laura Dombro (20th century)
“Men can intoxicate themselves with ideas as effectually as with alcohol or with bang and produce, be dint of serious thinking, mental conditions hardly distinguishable from monomania.”
—Thomas Henry Huxley (1825–95)