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

Put $p=\gamma (0)$, $v=\dot{\gamma }(0)$, $a=J(0)$ and $b=D_{t}J(0)$. Choose the short curve $c(s)={\exp }_p(sa)$ and let $P_s:T_{p}M\to T_{c(s)}M$ be parallel transport along $c$. The tangent vectorssatisfy $V(0)=v$ and ${\nabla }_{s}V(0)=b$. Define a geodesic variation byFor each fixed $s$ this is an affinely parametrized geodesic with initial point $c(s)$ and initial velocity $V(s)$. Although the Riemannian manifold need not be complete, smooth dependence for the geodesic equation on a neighborhood of the compact reference segment provides a single $\epsilon >0$ on which $F$ exists for all $|s|<\epsilon$ and $0\le t\le 1$. Thus completeness is unnecessary.

Set $X(t)={\partial }_{s}F(0,t)$. The torsion-free Levi-Civita connection gives ${\nabla }_s{\partial }_{t}F={\nabla }_t{\partial }_{s}F$. Differentiating ${\nabla }_t{\partial }_{t}F=0$ with respect to $s$, and commuting the two covariant derivatives, yieldsAlso $X(0)=c^{\prime }(0)=a$ and $D_{t}X(0)={\nabla }_{s}V(0)=b$. Uniqueness for the Jacobi field equation now gives $X(t)=J(t)$ on the whole segment. This is realization of Jacobi fields by geodesic variations.

The printed formula switches the order of its two variables. The consistent notation here is $F(0,t)=\gamma (t)$ and $J(t)={\partial }_{s}F(0,t)$. A parametrized surface here means a smooth variation map; it cannot be required to be an immersion, since the allowed Jacobi field $J=0$ would already violate that condition along the reference curve.