Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 7 / 5 / a

The Gauss lemma says that, for $v,w\in T_{p}M$ for which the exponential map is defined,In particular radial and angular directions are orthogonal, and the radial direction retains its length.

Take the variation through geodesics $\alpha (t,s)={\exp }_p(t(v+sw))$, and put $T={\partial }_t\alpha$, $J={\partial }_s\alpha$. The Levi-Civita connection is metric compatible and torsion-free, so ${\nabla }_{t}J={\nabla }_{s}T$. Each $t$-curve is a geodesic, hence ${\nabla }_{t}T=0$, and its constant squared speed is $|v+sw{|}_p^2$. ThereforeAt $t=0$ the variation starts at the fixed point $p$, so $J(0,s)=0$. Integrating from $0$ to $1$ and setting $s=0$ gives the displayed identity, because $T(1,0)=(d{\exp }_p{)}_{v}v$ and $J(1,0)=(d{\exp }_p{)}_{v}w$.

In geodesic normal coordinates based on an orthonormal basis at $p$, choose $v=x$ and $w$ to be the $j$th coordinate vector. The lemma givesThese identities also hold at $x=0$ by continuity. They are the radial identities required in part (b).