A vanishing theorem on manifolds of positive spectrum (Q1882366)

Starting from some results of Li-Wang and Leung-Wan, the author studies complete Riemannian manifolds \(M\) of dimension \(\geq\) 3. Suppose that the lower bound of the spectrum of the Laplacian is \(\lambda_1(M) > 0\) and that \(\text{Ric}_M \geq -(n-1)\lambda_1(M) (n-2)^{-1}\). Then, the author proves that M splits as a warped product of the real line and a compact manifold, or any smooth map from \(M\) to a manifold of non-positive sectional curvature, which is constant outside a compact set, is homotopic to a constant map.