= Solution
The <Gauss lemma> says that, for $v,w\in T_pM$ for which the <exponential map> is defined,
$$
\boxed{\left\langle(d\exp_p)_v v,(d\exp_p)_v w\right\rangle_{\exp_p(v)}=\langle v,w\rangle_p.}
$$
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 $\nabla_tJ=\nabla_sT$. Each $t$-curve is a <geodesic>, hence $\nabla_tT=0$, and its constant squared speed is $|v+sw|_p^2$. Therefore
$$
\frac{d}{dt}\langle T,J\rangle
=\langle T,\nabla_tJ\rangle
=\langle T,\nabla_sT\rangle
=\frac12\partial_s|T|^2
=\langle v+sw,w\rangle_p.
$$
At $t=0$ 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 gives
$$
g_{ij}(x)x^i=x_j,\qquad\boxed{g^{ij}(x)x_i=x_j}.
$$
These identities also hold at $x=0$ by continuity. They are the radial identities required in part (b).
Back to article page
