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.
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Christian Bär
University of Potsdam
Rudolf Zeidler
University of Potsdam
Comptes Rendus Mathématique
University of Potsdam
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Bär et al. (Tue,) studied this question.
synapsesocial.com/papers/69bb9321496e729e62980fea — DOI: https://doi.org/10.5802/crmath.825