  This paper concerns obstruction flatness of hypersurfaces $\Sigma$ that arise
as unit sphere bundles $S(E)$ of Griffiths negative Hermitian vector bundles
$(E, h)$ over K\"ahler manifolds $(M, g).$ We prove that if the curvature of
$(E, h)$ satisfies a splitting condition and $(M,g)$ has constant Ricci
eigenvalues, then $S(E)$ is obstruction flat. If, in addition, all these
eigenvalues are strictly less than one and $(M,g)$ is complete, then we show
that the corresponding ball bundle admits a complete K\"ahler-Einstein metric.
