Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 24H / b / Solution

A parametrized curve is a geodesic when its covariant acceleration vanishes, ${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$, for the induced Levi-Civita connection. In coordinates this isEquivalently, for an embedded surface its ordinary acceleration is normal to the tangent plane. The parameter is affine; arbitrary reparametrizations need not remain geodesics.

The local existence and uniqueness theorem says that for every $p\in X$ and $v\in T_{p}X$ there is a unique geodesic with $\gamma (0)=p$, $\dot{\gamma }(0)=v$, on a maximal open interval containing zero, depending smoothly on $(p,v)$. This follows from the local existence and uniqueness theorem for the smooth first-order system in $(\gamma ,\dot{\gamma })$. The exponential map is ${\exp }_p(v)={\gamma }_v(1)$ wherever the solution exists through time one. Its domain contains a neighborhood of zero, and $d({\exp }_p{)}_0=I$, since ${\exp }_p(tv)={\gamma }_v(t)$. Thus it is a local diffeomorphism near zero. Completeness is not assumed for a general surface; on a complete surface it is defined on all of $T_{p}X$.