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Geometric Properties of the Ferrand Metric

Julian, Poranee K.
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1353088820

Abstract Details

2012, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
We investigate the differentiability of the Ferrand metric-density and find a sufficient condition that implies that the Ferrand metric-density is not differentiable and that the Ferrand Gaussian curvature is negative infinity at a point. We prove that a closed Euclidean ball, closed half-space, or the complement of an open Euclidean ball contained in a domain is convex with respect to the Ferrand metric. As a result, the arc-length parametrization of a Ferrand geodesic has Lipschitz continuous first derivative, and this result is sharp. We consider the Ferrand geometry of a convex domain. We show that Ferrand geodesics are unique, that Ferrand balls are strictly Euclidean convex, and that Ferrand geodesics can be prolonged to Ferrand geodesic rays. Finally, we study the notion of a rolling disk condition and provide a sufficient geometric condition which guarantees that the ratio of the hyperbolic and quasihyperbolic metric-densities, and also the ratio of the Ferrand and quasihyperbolic metric-densities, both approach one as an interior point approaches a boundary point.
David Herron, Ph.D. (Committee Chair)
David Freeman, Ph.D. (Committee Member)
Carl David Minda, Ph.D. (Committee Member)
Nageswari Shanmugalingam, Ph.D. (Committee Member)
130 p.
Mathematics
Ferrand; conformal metric; geodesic; Mobius invariant; geometry; hyperbolic;

Recommended Citations

Citations

  • Julian, P. K. (2012). Geometric Properties of the Ferrand Metric [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1353088820

    APA Style (7th edition)

  • Julian, Poranee. Geometric Properties of the Ferrand Metric. 2012. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1353088820.

    MLA Style (8th edition)

  • Julian, Poranee. "Geometric Properties of the Ferrand Metric." Doctoral dissertation, University of Cincinnati, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1353088820

    Chicago Manual of Style (17th edition)

ucin1353088820
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© 2012, all rights reserved.
This open access ETD is published by University of Cincinnati and OhioLINK.