  It is shown that existence of a global solution to a particular nonlinear
system of second order partial differential equations on a complete connected
Riemannian manifold has topological and geometric implications and that in the
domain of positivity of such solution its reciprocal is the radial function of
only one of the following rotationally symmetric hypersurfaces in $\Rn$:
paraboloid, ellipsoid, one sheet of a two-sheeted hyperboloid, and a
hyperplane.
