  In Gromov's treatise Partial Differential Relations (volume 9 of Ergebnisse
der Mathematik und ihrer Grenzgebiete (3), 1986), a continuous map between
Riemannian manifolds is called isometric if it preserves the length of
rectifiable curves. In this note we develop a method using the Baire category
theorem for constructing such isometries. We show that a typical $1$-Lipschitz
map is isometric in canonically formulated extension and restriction problems.
