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 \(\mathbb{S}^{n}\) $${\mathbb{S}}^n$$ 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 \(\mathbb{S}^{n}\) $${\mathbb{S}}^n$$ is replaced by an arbitrary smooth strictly convex closed hypersurface in ℝⁿ⁺¹. The results answer two questions by Gromov (2023).