Dmitri Yu Burago, Dissertation Advisor/Co-Advisor Dmitri Yu Burago, Committee Chair/Co-Chair Augustin Banyaga, Committee Member Mark Levi, Committee Member Abhay Vasant Ashtekar, Committee Member
Keywords:
dynamical systems Riemannian geometry geodesics
Abstract:
This dissertation is divided into two parts. In part one we deal with the 1/k length
spectrum of a compact Riemannian manifold. The 1/k spectrum was introduced
by C. Sormani and has many relations with other geometrical objects. We will
show that there exists a class of manifolds with empty 1/k length spectrum. In
part two we work on the security of a manifold. A compact Riemannian manifold is
said to be uniformly secure if there is a number n in N such that for any two points
the set of geodesics connecting them can be blocked by n point obstacles. A general
conjecture is that uniform security implies flatness. We will prove this conjecture
for non-simply connected, orientable, two dimensional Riemannian manifolds.
