Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 16 / 1 / b / Solution

Suppose $J$ is a Jacobi field vanishing at both endpoints of a nonconstant geodesic segment $[0,t_0]$. Its tangential component $h(t)=⟨J(t),T(t)⟩$ satisfiesbecause $D_{t}T=0$ and the Riemann curvature tensor has the usual skew-adjoint symmetry. The two zero endpoint values imply $h\equiv 0$. Both $J$ and $D_{t}J$ are therefore perpendicular to $T$.

The linear map from these Jacobi fields to $T(0{)}^{\perp }$ given by $J\mapsto D_{t}J(0)$ is injective: $J(0)=0$ together with $D_{t}J(0)=0$ forces $J\equiv 0$ by uniqueness for the linear ordinary differential equation defining a Jacobi field. Its target has dimension $n-1$, soThis proves the endpoint-vanishing normal Jacobi fields have dimension at most n-1 bound. For a constant geodesic the equation is $J^{\prime \prime }=0$, and two zero values instead force $J=0$, so there is no conjugate point.