Suppose $M$ is a complete Riemannian manifold.Given two points $p$,$q$, there is a minimizing geodesic $\gamma$ connecting $p$and $q$ and the length of $\gamma=d(p,q)$. My question is, if a piecewise differentiable curve $c$ connecting $p$ and $q$ has length $d(p,q)$, can we show $c$ is differentiable?
2026-03-28 13:59:10.1774706350
Minimizing curve is differentiable
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Yes, length minimizing curves are (as) smooth (as the amibient manifold). This is (basically) true because a length minimizing satisfies a differential equation (the geodesic equation, which is the variational equation for the length functional), and it is known that solutions to this equation are smooth.
You should consult a textbook on differential geometries if you want to known the details. By googling I just found this reference, which might help you out.