Gromov-Lawson nonexistence theorem

This article describes a result related to the sectional curvature of a Riemannian manifold

Statement

A compact manifold which carries a Riemannian metric of everywhere nonpositive sectional curvature, cannot admit a metric with everywhere nonnegative scalar curvature, other than the flat metric.

References

• Positive scalar curvature and the Dirac operator on complete Riemannian manifolds by Mikhail Gromov and H. Blaine Lawson, Jr., Publ. Math. IHES, 58 (1963), 83-196