Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 24H / ii

The Gauss lemma says that the radial and angular directions of the exponential map are orthogonal, with radial lengths preserved. On a surface it givesso the metric in geodesic polar coordinates is $d{r}^2+G(r,\theta )d{\theta }^2$.

To prove it, consider a smooth variation $\gamma (r,s)={\exp }_p(rv(s))$ of geodesics with unit initial vectors $v(s)$. Put $T={\gamma }_r$ and $J={\gamma }_s$. Each radial geodesic has constant unit speed, and the Levi-Civita connection is torsion free. HenceAt $r=0$, $J=0$ because all geodesics start at $p$, so the constant inner product is zero. Unit radial length follows from constant speed. General initial-vector variations split into a radial component, whose image has the same length, and a component tangent to the initial sphere, whose image is orthogonal to it. This proves the full Gauss lemma.