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.

Author Bio

Maxine Elena Calle is a member of the Class of 2020 at Reed College in Portland, Oregon. She will receive her undergraduate degree in mathematics, with an additional concentration in philosophy. Her desire to pursue mathematical research was sparked by the Research Experience for Undergraduates (REU) program at California State University, San Bernardino, where this research was completed. She plans to go on to get a Ph.D in mathematics and pursue a research career. Maxine’s life is balanced by a variety of other interests, including gardening, music performance, and aerial acrobatics.