Commentarii Mathematici Helvetici

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Volume 90, Issue 4, 2015, pp. 761–798
DOI: 10.4171/CMH/370

Published online: 2015-12-03

Equidimensional isometric maps

Bernd Kirchheim[1], Emanuele Spadaro[2] and László Székelyhidi Jr.[3]

(1) Universität Leipzig, Germany
(2) Max-Planck-Institut Mathematik in den Naturwissenschaften, Leipzig, Germany
(3) Universität Leipzig, Germany

In Gromov’s treatise (Partial diff erential 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.

Keywords: Isometric maps, fine differential inclusions, Baire category method

Kirchheim Bernd, Spadaro Emanuele, Székelyhidi Jr. László: Equidimensional isometric maps. Comment. Math. Helv. 90 (2015), 761-798. doi: 10.4171/CMH/370