This research generalizes the two invariants known as constant sectional curvature (csc) and constant vector curvature (cvc). We use k-plane scalar curvature to investigate the higher-dimensional analogues of these curvature conditions in Riemannian spaces of arbitrary finite dimension. Many of our results coincide with the known features of the classical k=2 case. We show that a space with constant k-plane scalar curvature has a uniquely determined tensor and that a tensor can be recovered from its k-plane scalar curvature measurements. Through two example spaces with canonical tensors, we demonstrate a method for determining constant k-plane vector curvature values, as well as the possibility of a connected set of values. We also generate loose bounds for candidate values based on sectional curvatures. By studying these k-plane curvature invariants, we can further characterize model spaces by generating basis-independent numbers for various subspaces.
Dr. Corey Dunn
Calle, Maxine E.
"k-Plane Constant Curvature Conditions,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 20
, Article 6.
Available at: https://scholar.rose-hulman.edu/rhumj/vol20/iss2/6