How rapidly do nearby geodesics separate?
10.1. Geodesics and Jacobi Fields
10.1a. Vector Fields Along a Surface in Mn
Let x:U ⊂ 2 → Mn be a differentiable map of a rectangle in the plane into Mn. We call this map a (parameterized) surface even though we put no demands on the rank of the differential x*; that is, ∂x/∂u1 and ∂x/∂u2 may be dependent.
Let us again put u1 = u and u2 = ν.
A smooth map v: U ⊂ 2 → M that ...