On the principal Ricci curvatures of a Riemannian 3-manifold

Milnor has shown that three-dimensional Lie groups with left invariant Riemannian metrics furnish examples of 3-manifolds with principal Ricci curvatures of fixed signature\,---\,\emph{except} for the signatures $(-,+,+)$, $(0,+,-)$, and $(0,+,+)$. We examine these three cases on a Riemannian 3-manifold, and prove the following. If the manifold is closed, then the signature $(-,+,+)$ is not globally possible if it is of the form $-\mu,f,f$, with $\mu$ a positive constant and $f$ a smooth function that never takes the values $0,-\mu$ (hence this also applies to the signature $(-,-,-)$). In the scalar flat and complete case, the signature $(0,+,-)$ is not globally possible if the eigenvalues are constants and the zero eigenspace is spanned by a unit length vector field with geodesic flow; if the manifold is closed and this vector field is also divergence-free, then $(0,+,-)$ is not possible even if the nonzero eigenvalues are not constant. Finally, on a connected and complete Riemannian 3-manifold, if $(0,+,+)$ occurs globally and the two positive eigenvalues are equal, then the universal cover splits isometrically.