Myers-type theorems and some related oscillation results (Q692342)

The authors investigate the differential equation \( u''(t)+K_{\gamma}(t) u(t)=0\) along a geodesic \(\gamma\) where the function \(K_{\gamma}(t)\) is defined using the Ricci curvature along \(\gamma,\) i.e. \(K_{\gamma}= \text{Ric}(\gamma', \gamma')/(m-1)\). They present sufficient conditions on the Ricci curvature to ensure the existence of a first zero of a solution \(u(t)\) at finite time. As a consequence, one obtains a generalization of the classical Myers theorem, namely sufficient conditions for the compactness of a complete Riemannian manifold in terms of integral bounds for the Ricci curvature along a minimal geodesic