Constant Vector Curvature in Three Dimensions

Differential geometry is the use of the techniques and tools of calculus to study the geometric properties of manifolds. One of the most commonly studied properties of manifolds their curvature. We can measure the curvature of a manifold at a point by using a metric called an algebraic curvature tensor and a geometric object known as a model space. A model space is formed when a manifold, inner product, and algebraic curvature tensor are grouped together. There are several curvature conditions that a model space can satisfy. This research is concerned with the necessary and sufficient conditions for a model space in three dimensions with positive definite inner product to have the specific curvature condition of constant vector curvature. This presentation summarizes the background for this research along with its findings.