2014 Student Research Conference:
27th Annual Student Research Conference

On the Logical Inequivalence of Geodesic Completeness in semi-Riemannian Manifolds
Garrett R. Yord
Dr. Mohammad Samiullah, Faculty Mentor

The study of geodesic completeness is a dominant theme in Riemannian geometry, and there are a number of well-known results that give necessary and sufficient conditions for a Riemannian manifold to be geodesically complete. The Hopf-Rinow Theorem is generally regarded as one of the most powerful results in this area. In the arena of the Lorentzian manifolds of general relativity, however, the Hopf-Rinow theorem has no analogue. Indeed, every single equivalence given by the Hopf-Rinow theorem for Riemannian manifolds fails to hold for semi-Riemannian manifolds. As a consequence, the three types of geodesic completeness studied in Lorentzian manifolds (timelike, spacelike, and null geodesic completeness) are logically unrelated. I propose to prove that the three types of geodesic completeness are logically unrelated by constructing examples which show a Lorentzian manifold may be complete in any two of these categories while being incomplete in the third.

Keywords: semi-Riemannian geometry, geodesic completeness


Presentation Type: Oral Paper

Session: 102-4
Location: MG 1096
Time: 8:45

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