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Abstract

This research investigates a model space invariant known as k-plane constant vector curvature, traditionally studied when k=2, and introduces a new invariant, (m,k)-plane constant vector curvature. We prove that the sets of k-plane and (m,k)-plane constant vector curvature values are connected, compact subsets of the real numbers and establish several relationships between the curvature values of a decomposable model space and its component spaces. We also prove that every decomposable model space with a positive-definite inner product has k-plane constant vector curvature for some integer k>1. In two examples, we provide the first instance of a model space with (m,k)-plane constant vector curvature and leverage our theorems to efficiently calculate the k-plane constant vector curvature values of a decomposable model space. This research further characterizes model spaces by assigning new basis-independent values to its various subspaces and allows us to easily construct model spaces with prescribed curvature values.

Author Bio

Kevin Tully is a 2020 graduate of Wheaton College in Illinois. His participation in the Budapest Semesters in Mathematics program and the Research Experience for Undergraduates at California State University, San Bernardino, where this work was completed, sparked his interest in differential geometry. He is currently pursuing a PhD in pure math at the University of Washington, Seattle. When not studying, he enjoys watching and playing sports and spending time with his family’s five pets.

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