Undergraduate Honors Theses

Thesis Defended

Spring 2013

Document Type

Thesis

Department

Mathematics

First Advisor

Prof. Jeanne Clelland

Abstract

The primary defining characteristic of Euclidean geometry in R3 is the presence of a flat metric, h,i which is defined on all tangent vectors to all points in R3 and invariant under the action of the Euclidean group. When studying submanifolds of the Euclidean space E3 (i.e., R3 together with a Euclidean metric), all metric properties (e.g., arc lengths and surface areas) are derived from this underlying metric. By contrast, in equiaffine geometry (which, for convenience, we will refer to simply as “affine geometry”), it is not possible to define a metric on tangent vectors which is preserved by the action of the equiaffine group. There is an invariant volume form, but no invariant notion of distance which can be restricted to submanifolds of A3 (i.e., R3 together with an equiaffine structure, which we will define shortly) in any obvious way. Nevertheless, it is possible to define a notion of affine metric for generic surfaces in such a way that this notion is preserved by the action of the equiaffine group. Because there is no inner product on tangent vectors, this affine notion of metric on submanifolds depends on higher-order derivatives, as opposed to the analogous Euclidean notion, which depends only on the first derivatives of a surface. Once we construct a measure of distance on an affine surface, we implicitly construct a notion of curvature on a surface–i.e., how that surface bends and changes. From this, the notions of affine minimal surfaces and affine flat surfaces emerge [1]. An affinely flat surface is a surface with zero affine curvature. An affinely minimal surface is the affine analogue to a Euclidean minimal surface—a surface that has locally extremal surface area. In this work, we characterize hyperbolic1 affine surfaces that are both affine minimal and affine flat.

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