Tubular neighbourhoods of 2-flats in 4-manifolds of nonpositive curvature. (Q5948673)

Let \(M\) be a closed, connected Riemannian manifold of nonpositive sectional curvature. A closed flat in \(M\) is a totally geodesic, isometrically immersed \(k\)-dimensional Euclidean torus, where \(k\geq 2\). In this paper, the author proves that every orientable \(\mathbb{R}^2\)-vector bundle over the \(2\)-torus arises as a tubular neighbourhood of a \(2\)-flat in a closed \(4\)-manifold of nonpositive sectional curvature and rank one.