MATH Seminar
| Title: Rotationally Symmetric Planes in Comparison Geometry |
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| Defense: Dissertation |
| Speaker: Eric Choi of Emory University |
| Contact: Eric Choi, echoi7@emory.edu |
| Date: 2012-11-08 at 12:00PM |
| Venue: MSC W502 |
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| Abstract: Global Riemannian geometry is concerned with relating geometric data such as curvature, volume, and radius to topological data. Cheeger-Gromoll showed that any noncompact complete manifold $M$ with nonnegative sectional curvature contains a boundaryless, totally convex, compact submanifold $S$, called a soul, such that $M$ is homeomorphic to the normal bundle over $S$. In the first part of our talk, we show that if $M$ is a rotationally symmetric plane $M_m,$ defined by metric $dr^2 + m^2(r) d\theta^2$, then the set of souls is a closed geometric ball centered at the origin, and if furthermore $M_m$ is a von Mangoldt plane, then the radius of this ball can be explicitly determined. We show that the set of critical points of infinity in $M_m$ is equal to this set of souls, and we make some additional observations on the set of critical points of infinity when $M_m$ is von Mangoldt with a point at which the sectional curvature is negative. We close out the first part of the talk with showing that the slope $m^{\prime}(r)$ of $M_m$ near infinity can be prescribed with any number in $(0, 1]$. In the second part of our talk, we extend results in a paper by Kondo-Tanaka in which the authors generalize the Toponogov Comparison Theorem such that an arbitrary noncompact manifold $M$ is compared with a rotationally symmetric plane $M_m$ satisfying certain conditions to establish that $M$ is topologically finite. We substitute one of the conditions for $M_m$ with a weaker condition and show that our method using this weaker condition enables us to draw further conclusions on the topology of $M$. We also completely remove one of the conditions required for the Sector Theorem, one of the principal results in the same paper. |
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