Llarull’s scalar curvature rigidity theorem states that a 1 -Lipschitz map f : M → 𝕊 n from a closed connected Riemannian spin manifold M with scalar curvature scal ≥ n ( n - 1 ) to the standard sphere 𝕊 n is an isometry if the degree of f is nonzero. We investigate if one can replace the condition deg ( f ) ≠ 0 by the weaker condition that f is surjective. The answer turns out to be “no” for n ≥ 3 but “yes” for n = 2 . If we replace the scalar curvature by Ricci curvature, the answer is “yes”in all dimensions.
Bär et al. (2026) studied this question.