Let \((M,g)\) be a compact Riemannian manifold of \(\dim M \geq 11\). Theorem: If \(g\) is either negatively curved or if \((M,g)\) is a flat torus, then there is a Riemannian metric \(h\) on \(M\) and a harmonic homotopy equivalence \(f : (M,h) \to (M,g)\); yet \(f\) is not a homeomorphism. Theorem: If \(M\) supports a nonpositively curved metric, then it carries a diffeomorphism homotopic but not isotopic to the identity. The proofs rely on earlier work of the authors [Proc. Symp. Pure Math. 54, Part 3, 229-274 (1993; Zbl 0796.53043)].