2 November 2015
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 --- except for the signatures (-,+,+), (0,+,-), and (0,+,+). We examine these three cases on a Riemannian 3-manifold, and prove global obstructions in certain cases. For example, if the manifold is closed, then the signature (-,+,+) is not globally possible if it is of the form -µ,f,f, with µ a positive constant and f a smooth function that never takes the values 0,-µ (this generalizes a result by Yamato '91). Similar obstructions for the other cases will also be discussed. Our methods of proof rely upon frame techniques inspired by the Newman-Penrose formalism. Thus, we will close by turning our attention to four dimensions and Lorentzian geometry, to uncover a relation between null vector fields and exact symplectic forms, with relations to Weinstein structures.
- Geometry and Analysis Seminar