Boundary convexity on manifolds with nonnegative Ricci curvature.
Let \(M^n\) be an \(n\)-dimensional compact connected Riemannian manifold with non-empty boundary \(\partial M\), and let \(S_{\partial M}: X\to \nabla_X\mu\) be the second fundamental form on \(\partial M\) with respect to the unit outward normal \(\mu\). For \(\lambda=\inf\) of eigenvalues of \(S_{\partial M}\) over \(\partial M\), if \(\lambda> 0\) then \(\partial M\) is convex, and convexity or strict convexity imply conditions on \(\lambda\) (with \(\lambda\geq 0\)). If \(n= 3\), a Ricci-positively curved \(M^3\) with \(\lambda> 0\) is diffeomorphic to the 3-dimensional ball, as the author shows in the Appendix. For \(n\geq 4\), there are manifolds \(M\) with positive Ricci curvature and convex \(\partial M\) that are not contractible. So a stronger convexity condition is needed for contractibility. Let \(\Lambda(M^n)= \lambda({\text{vol}(\partial M)\over \omega_{n-1}})^{{1\over n-1}}\), \(\omega_{n-1}=\text{volume}\) of the unit \((n-1)\)-sphere, \(\text{Ric}_M>0\), \(K_M\geq 0\) at \(\partial M\), and assume \(\exists \delta_n: 1> \delta_n> 0\) with \(\Lambda(M)> 1-\delta_n\), then \(M\) is contractible.
scientific article; zbMATH DE number 1463737