Surfaces and singularities
|
| Mikhail Malakhaltsev |
| Universidad de los Andes |
| Abstract |
| The classical geometry of surfaces in the Euclidean space \(\mathbb{R}^3\) is dedicated to the study of regular surfaces, which are subsets of \(\mathbb{R}^3\) that are locally homeomorphic to the plane and have tangent planes at all points. For example, the solutions to the equation \(F(x, y, z) = c\), where the gradient \(\nabla F\) does not vanish, are regular surfaces. |
| However, a question arises in relation to the so-called {\em singular points}, where vanish the gradient of \(F\). How do solutions behave in a neighborhood of a singular point? |
| In fact, singular points appear naturally in various studies on the geometry of surface. These points are found, for example, in the projections of surfaces on a plane, in the envelopes of families of curves or surfaces, and in the description of the points where the Gaussian curvature of the surface cancels out. |
| To address these issues related to singular points, the theory of sin- gularities of smooth application is used. This discipline was developed mainly between the 1960s and 1980s and currently has numerous applications in science and engineering. In this context, it is sometimes also called ``catastrophe theory". |
| In this short course, we will explore the essential concepts of singularity theory with a particular focus on its application to the geometry of surfaces in Euclidean space. |
| Content: |
- What is a surface?
- What is a singularity?
- Singularities of applications of \(\mathbb{R}^n\to \mathbb{R}\).
- Morse’s Lemma.
- Implicitly given surfaces.
- Singularities of applications of \(\mathbb{R}^2\to \mathbb{R}\).
- Whitney’s theorem.
- Singularities of surface projections.
- Envelopes of families of curves.
- Singularities of applications of \(\mathbb{R}\to \mathbb{R}\).
- Deformations (families) of applications and singularities.
- Singularities of the Gauss map, points of zero curvature, \(G\)-structures and umbilical points.
- Zeros of vector fields and Euler characteristic.
|
| Bibliography. |
| 1. V.I.Arnold. \emph{The theory of singularities and its applications}, 1991. |
| 2. J.W. Bruce, P.J. Giblin \emph{Curves and singularities}, 1984. |
