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Pure Math Graduate Student Math Seminar: Andrew Shedlock, A smooth two Parameter Family of Geodesically Equivalent Metrics
February 20 | 2:00 pm - 3:00 pm EST
In Riemannian geometry, given a Riemannian manifold (M,g) one can use geodesics associated with (M,g) to determine information about the shortest distance between points, curvature, triangles on a manifold and Euler characteristic of the space of M in special cases. Thankfully, given a metric on a manifold, we can always determine geodesics on said manifold.
A partial converse to this question is: what can one learn about a Riemannian manifold if they know the paths of all of the geodesics? To answer this question, the notion of geodesically equivalent metrics was developed. We say that two Riemannian manifolds (M, g1) and (M,g2) are geodesically equivalent if their geodesics look the same. In this talk I describe a 2 parameter family of Riemannian metrics on the closed unit ball in R^n which are geodesically equivalent to the standard Euclidean metric (i.e. straight lines) and describe geometric properties of each metric in this family.
Zoom Meeting link: https://ncsu.zoom.us/j/92762214990?pwd=MnA1TnNVQUpzUlo3cTM5RmlNWVF4Zz09