Past exam of the geometry course of the University of Oxford / 2008 / A / 3

For the unit-speed curve, the normal curvature is ${\kappa }_n=t^{\prime }\cdot n$, and the geodesic curvature is ${\kappa }_g=t^{\prime }\cdot (n\times t)$. Differentiating $\gamma =r(u,v)$ twice and taking the normal component kills the terms in $r_u,r_v$, givingAn affinely parametrized geodesic has zero tangential acceleration, or equivalently its covariant derivative of velocity is zero. For unit speed, $t^{\prime }\cdot t=0$, and $t,n\times t,n$ form an orthonormal frame. Therefore $t^{\prime }={\kappa }_{n}n+{\kappa }_g(n\times t)$, so zero tangential acceleration is exactly ${\kappa }_g=0$.

Since $E{u}^{\prime }+F{v}^{\prime }={\gamma }^{\prime }\cdot r_u$, differentiation givesThe corresponding calculation with ${\gamma }^{\prime }\cdot r_v=F{u}^{\prime }+G{v}^{\prime }$ givesThus the two requested equations hold exactly when ${\gamma }^{\prime \prime }$ is perpendicular to both tangent basis vectors, which is precisely the geodesic condition.

On expanding the left sides, the coefficient matrix of $(u^{\prime \prime },v^{\prime \prime })$ is the positive-definite metric matrix. It is invertible, so the equations form a smooth second-order ordinary differential equation in local coordinates. The local existence and uniqueness theorem gives a unique solution for each initial point and coordinate velocity. For an arbitrary initial tangent vector $w$ this is an affine parameter of constant speed $|w|$; it is arclength when $|w|=1$. For $w=0$ the solution is the constant curve. This specifies how the arbitrary-velocity statement relates to the preceding unit-speed convention.

Let $R$ be reflection in the plane. Choose a point of the regular intersection curve and a unit tangent vector $w$ to it. Both the point and $w$ are fixed by $R$. The unique local geodesic with these initial data is carried by the surface isometry $R$ to another geodesic with the same initial data. Uniqueness therefore gives $R\gamma =\gamma$. Its image lies in the fixed-point set of the reflection, namely the plane, hence locally in the intersection curve. Thus the intersection curve is a geodesic, when parametrized by arclength.