= A minimizing broken geodesic has no corner
At an interior junction of two unit-speed <geodesic> segments, let $u$ be the incoming and $v$ the outgoing tangent. Choose points close to the junction in a common <convex normal neighbourhood>. Moving the junction with velocity $z$ changes the sum of the two segment lengths by $\langle u-v,z\rangle$, using the <first variation of geodesic energy> or the <Gauss lemma>. If $u\ne v$, choose $z=v-u$ to reduce length. Hence a length-minimizing broken <geodesic> has matching tangents and is one smooth <geodesic>.
// Target: riemannian-geometry.bigb
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