Rectifiable families of conics.
Published: 2004
Language: eng
Number of pages: 92
Description: In dimension 2, I have a complete description of sufficiently large rectifiable pencils of conics (Chapter 4). Conics from such pencil are the images of lines under some fractional quadratic map. Geometrically, almost all conics from a generic rectifiable pencil are characterized by the property that they have 3 points of tangency with the same curve dual to a cubic.Rectifiable pencils of curves appear, for example, in Riemannian geometry---geodesics passing through a given point are rectifiable by the inverse of the exponential map. More generally, E. Cartan's normal projective connections describe exactly point-wise rectifiable systems of curves. The study of such systems was motivated by the General Theory of Relativity. Many distinguished geometers of the past, including Blaschke, Cartan, Veblen and Thomas, worked on the subject.A set of curves with a common point (also called a  pencil of curves) will be called rectifiable if there is a local diffeomorphism that maps all curves from this set to straight lines.In dimensions 2 and 3, a description of all sufficiently large rectifiable pencils of circles was known (A. Khovanskii and F. Izadi): all circles that form such pencils must share a second common point. It turns out that this is no longer true in dimension 4. However, I give a complete answer (Chapter 2), which is related to the quaternionic algebra (quaternionic Hopf fibrations). This answer gives rise to the following results. If a smooth diffeomorphism between an open subset of the 4-dimensional projective space and an open subset of the 4-dimensional sphere takes germs of lines to germs of circles, then it comes from a quaternionic Hopf fibration or from a classical model of a classical geometry. If all real geodesics of a Kahler metric on a part of the 2-dimensional complex space are arcs of circles, then this metric is a Fubini metric (a classical model of a space with constant holomorphic sectional curvature).In some special cases, an explicit description of rectifiable pencils of circles is given in terms of representations of Clifford algebras (Chapter 3).In any dimension, I reduce the description of sufficiently large rectifiable pencils of circles to the description of quadratic forms between spheres (Chapter 1). Quadratic forms between spheres were studied in relation with the Hurwitz problem about sums of squares (Hurwitz, 1898) and homotopy groups of spheres.In this thesis, I study rectifiable pencils of very simple algebraic curves---circles and, more generally, conics. Rectification problems for circles are motivated by nomography.
Notes: Adviser:  A. Khovanskii.

Thesis (Ph.D.)--University of Toronto, 2004.

Electronic version licensed for access by U. of T. users.

Source: Dissertation Abstracts International, Volume: 65-10, Section: B, page: 5184.