On hypersurfaces in a Riemannian vector bundle with prescribed Gaussian curvature and convexity (Q1434206)

Let \(E\rightarrow M\) be a metric vector bundle over a Riemannian manifold \(M\) and \(\Sigma \subset E\) the unit sphere subbundle. The total space \(E\) is endowed with a compatible Riemannian metric such that the orthogonal complement to the vertical space is the horizontal space of a metric connection in the bundle \(E\rightarrow M\). The main results are: 1) there is no radial graph \(\Gamma \subset E\) over \(\Sigma\) of positive Gauss curvature, 2) the Gauss curvature of any convex radial graph is zero and the Gauss curvature of its fibres is constant along horizontal curves and 3) there exists a convex radial graph with fibres of prescribed positive Gauss curvature, under suitable assumptions on the prescribed function.