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Let M be a compact Riemannian manifold and E a Riemannian vector bundle on M.We look for hypersurfaces of E with a prescribed vertical Gaussian curvature.In trying to solve this problem fibre-wise, we loose the regularity of theresulting solution. To unsure the smoothness of the solution, we construct itas a radial graph over the unit sphere subbundle of E and prove its existenceby solving in this one a nonlinear partial differential equation ofMonge-Amp\`ere type.

On compact hypersurfaces in a Riemannian vector bundle with prescribed vertical Gaussian curvature