A smooth map with pointwise operator norm at most c is c lipschitz for riemannian distance

Given: The stated differential bound and two source points.

Riemannian distance on a connected manifold: On a connected Riemannian manifold define $d_g(p,q)=\inf \{L_g(\gamma ):\gamma \text{ is piecewise }{C}^1\text{ from }p\text{ to }q\}$. Lengths are those of def-riemannian-speed-and-length. For each pair $p,q$, lem-any-two-points-in-a-connected-smooth-manifold-can-be-joined-by-a-piecewise-c-one-curve supplies a curve, so the set of lengths is nonempty, contains a finite real number and is bounded below by zero. Applying the least-upper-bound property cor-cauchy-reals-lub-complete to the negatives gives a finite nonnegative infimum. On the empty connected manifold this defines the empty distance function; there are no pairs to evaluate. No minimizing curve is part of this definition.

Riemannian speed and length: The Riemannian speed on a $C^1$ piece is $\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_g=\sqrt{g_{\gamma (t)}(\dot{\gamma }(t),\dot{\gamma }(t))}$. Its length is $L_g(\gamma )={\sum }_j{\int }_{t_{j-1}}^{t_j}\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_{g}dt$. The curve convention is def-piecewise-c-one-curve-on-a-manifold and the norm is def-pointwise-norm-and-angle-from-a-riemannian-metric. Each integrand is continuous on its closed piece with the one-sided endpoint derivative, hence Riemann integrable and nonnegative. Values chosen at the finitely many corners do not change its integral. For a singleton interval the empty sum is zero; a constant curve also has zero length. Partition independence is established next.

Any two points in a connected smooth manifold can be joined by a piecewise c one curve: Any two points in a nonempty connected smooth manifold can be joined by a finite piecewise $C^1$ curve.