Area spaces: First steps
Let \(M\) be a compact Riemannian manifold. For a non-negative integer \(m\), let \(S_m(M)\) denote the abelian group of all \(m\)-dimensional Lipschitz chains with integer coefficients and \(B_m(M)\) the image of the boundary map \(\partial : S_{m+1}(M) \to S_m(M)\). The group \(B_1(M)\) is equipped with a semi-metric \(d(\gamma_1,\gamma_2) = \inf \{area(s): s \in S_2(M), \partial s=\gamma_1-\gamma_2 \}\). Let \(B(M)\) denote the associated metric space \(B_1(M)/d\), that is, we identify \( \gamma_1\) and \( \gamma_2\) if \( d(\gamma_1,\gamma_2) =0\). Let \( \overline B(M)\) be the completion of \(B(M)\). If \( \phi : M_1 \to M_2\) is a smooth map then \( \phi\) pushes forward curves and currents, and so induces a map \( \phi_* : \overline B(M_1) \to \overline B(M_2)\). The main result of the paper is the following: Theorem 1.3. Let \(M_1\) and \(M_2\) be compact Riemannian manifolds ( possibly with boundaries) of dimension \( \geqslant 3\). Suppose that the metric spaces \(\overline B(M_1)\) and \(\overline B(M_2)\) are isometric via a homogeneous isometry. (An isometry \(\Phi \) is homogeneous if \( \Phi(ks)=k \Phi (s)\) for all integer \(k\) ). Then \(M_1\) and \(M_2\) are isometric. Moreover, every homogeneous isometry \( \Phi : \overline B(M_1) \to \overline B(M_2)\) is induced by a Riemannian isometry \( \phi : M_1 \to M_2\) so that \( \phi_*=\Phi \). As shown by the authors, the theorem does not hold in dimension 2.
scientific article; zbMATH DE number 5652307