Curvature Group Summary

Monday - Friday, July 1 - 5, 2013

Our working group, the Curvature Crew, is attending the Undergraduate Summer School lectures on the Curvature of Space and Time, given by Iva Stavrov, Associate Professor of Mathematics at Lewis & Clark College. In addition, we spend an hour after each lecture debriefing concepts and working on problem sets with Brian Hopkins, SSTP staff.

In order to describe the curvature of space and time, we spent this week (and will continue next week) getting acquainted with spaces that are non-Euclidean. We began the week with specific examples to elucidate Riemann's Habilitation lecture. We then defined manifolds, which are spaces that look Euclidean when we consider small patches. Riemannian manifolds have an additional way to define distance, which may or may not be the same as how we measure Euclidean distance.

Armed with a rigorous definition of Riemannian manifolds, we then defined geodesics, which represent the shortest paths between any two points. In Euclidean space, this represents a straight line, but because Riemannian manifolds may have a different way to measure distance, straight lines are not necessarily geodesics. For example, on a sphere, geodesics are great circles, which are circles on the sphere that go through the center of the sphere. This is why a flight from Seattle to Paris will fly over Greenland instead of going across the 48th parallel.

During our debrief sessions, we try to clarify concepts and examples given during the lectures. We also work on selected problems from the problem sets.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.