Prof. Jeanne Clelland
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 . 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.
Miller, Jonah Maxwell, "A Characterization of Affine Minimal and Affine Flat Surfaces" (2013). Undergraduate Honors Theses. 444.