Spectral flow, Llarull's rigidity theorem in odd dimensions and its generalization

For a compact spin Riemannian manifold (M,g (TM)) of dimension n such that the associated scalar curvature k(TM )verifies that k (TM)>= n(n-1), Llarull's rigidity theorem says that any area-decreasing smooth map f from M to the unit sphere S-n of nonzero degree is an isometry. We present in this paper a new proof of Llarull's rigidity theorem in odd dimensions via a spectral flow argument. This approach also works for a generalization of Llarull's theorem when the sphere S-n is replaced by an arbitrary smooth strictly convex closed hypersurface in Rn+1. The results answer two questions by Gromov (2023).