Constant Vector Curvature in 3 Dimensions: A Complete Description

SMSU Event Center BC

A relatively new area of interest in differential geometry involves determining if a model space has the properties of constant vector curvature or constant sectional curvature. The natural setting in which to begin studying these properties is in 3-dimensional space. This paper in particular examines these properties in the Lorentzian setting, where all Ricci operators take on one of four Jordan-Normal forms. We determine that three of these four model space families (Ricci operators) possess the property of constant vector curvature, and that under an orthonormal basis, only the diagonalizable family has constant sectional curvature, and that is only when the Ricci Operator has precisely one eigenvalue. By examining these families together, we draw some interesting and unifying conclusions that may be useful for exploring these properties in higher dimensions.