Intersections of minimal submanifolds in manifolds of partially positive curvature (Q1906551)

The authors show that if \(N\) is a complete, connected \(n\)-dimensional Riemannian manifold with non-negative \(k\)-th Ricci curvature, if \(V\) is a complete, totally geodesic hypersurface, \(W\) an \(r(\geq k)\)-dimensional complete, minimal submanifold of \(N\) with both \(V\) and \(W\) immersed as closed subsets and if \(N\) has positive \(k\)-th Ricci curvature at all points of \(V\) or of \(W\), then \(V\) and \(W\) must intersect. The proof is based on the second variational formula for the arc-length of a geodesic formulated by \textit{J. L. Synge} [Proc. Lond. Math. Soc. 25, 247-264 (1926)]. This theorem is a higher-dimensional extension of a theorem of \textit{J. Hadamard} [J. Math. Pures Appl. 5, 331-387 (1897)] and a generalization of theorems of \textit{T. Frankel} [Pac. J. Math. 11, 165-174 (1961; Zbl 0107.39002)] and of \textit{S. B. Myers} [Trans. Am. Math. Soc. 25, 247-264 (1926)]. The case in which the ambient manifold \(N\) is a Kähler manifold is also considered. Applications to non-existence results for minimal immersions into a half space of a Riemannian manifold with partially positive curvature are given.