Curvature, Harnack's inequality, and a spectral characterization of nilmanifolds (Q1810770)

Generalizing a classical result of Bochner, which asserts that the \(n\)-torus is the only \(n\)-dimensional Riemannian manifold with positive semidefinite Ricci tensor and first Betti number equal to \(n\), the authors give sufficient conditions for a Riemannian manifold to be a nilmanifold or an infra-nilmanifold. Their hypothesis involves both bounds on the Ricci tensor for some integral norm and on the eigenvalues of the Laplacian acting on \(1\)-forms. The proof relies on some Harnack-type inequality.