= Solution
We use the following existence and uniqueness statement for <parallel transport>: for a smooth <affine connection>, a smooth curve $\gamma:[0,1]\to M$, an initial time $t_0$ and a vector $v\in T_{\gamma(t_0)}M$, there is a unique smooth <vector field along a map> $V$ on the whole curve satisfying $D_tV=0$ and $V(t_0)=v$. In a <frame of a vector bundle> this is the linear <ordinary differential equation> $\dot V=-\Gamma(\dot\gamma)V$. Its smooth coefficient matrix is bounded on every compact time subinterval in a trivializing neighborhood, and the linear existence theorem gives a solution throughout that subinterval. A finite subdivision of $[0,1]$ into such neighborhoods and uniqueness patch the solutions. No completeness hypothesis on $M$ is needed.
For arbitrary fields $U,V$ along a curve, the definition of the induced <covariant derivative> of the <metric tensor> gives
$$
\frac d{dt}g(U,V)
=(\nabla_{\dot\gamma}g)(U,V)+g(D_tU,V)+g(U,D_tV).
$$
This can also be checked by differentiating $g_{ij}(\gamma(t))U^i(t)V^j(t)$ and substituting the formula in part (a). If $\nabla$ is a <metric connection> and $V$ is parallel, it follows that $d(g(V,V))/dt=0$. Positivity of the <Riemannian metric> makes $|V|=\sqrt{g(V,V)}$ constant. Thus <parallel transport> preserves lengths.
Conversely assume every parallel field along every smooth curve has constant length. Fix $p\in M$ and arbitrary $x,v\in T_pM$. Choose a curve through $p$ with velocity $x$ at an interior time $t_0$. Explicitly, in a <coordinate chart> $z$ centered at $p$, take $\gamma(t)=z^{-1}(h(t)\,dz_p(x))$, where $h(t_0)=0$, $h'(t_0)=1$, and $h$ has sufficiently small amplitude and support near $t_0$. A <smooth bump function> produces such $h$, so the curve is defined on all of $[0,1]$ and remains inside the chart. By the stated <parallel transport> theorem there is a parallel field with $V(t_0)=v$. Evaluating the derivative of its squared length gives
$$
0=\left.\frac d{dt}g(V,V)\right|_{t_0}=(\nabla_xg)_p(v,v).
$$
For fixed $x$, the <bilinear form> $B_x(v,w)=(\nabla_xg)_p(v,w)$ is symmetric. Its diagonal values vanish, so <polarization identity> gives
$$
2B_x(v,w)=B_x(v+w,v+w)-B_x(v,v)-B_x(w,w)=0.
$$
Since $p,x,v,w$ were arbitrary, $\nabla g=0$. \b[A connection is metric-compatible if and only if all parallel fields have constant length.] Polarization also shows that its <parallel transport> preserves all <inner products>, not just lengths.
Back to article page
