Past exam of the geometry course of the University of Oxford / 2009 / A / 3 / i / Solution

For a regular parametrization $r(u,v)$, the first fundamental form is the induced metricIt measures the lengths and angles of tangent vectors. A geodesic is a curve whose unit tangent has zero covariant derivative along itself. For an arc-length parametrized surface curve in Euclidean three-space, this means its acceleration is normal to the surface, or ${\gamma }^{\prime \prime }\cdot r_u={\gamma }^{\prime \prime }\cdot r_v=0$.