Introduction to differential geometry of space curves and surfaces

Abstract

This thesis is an introduction to some of the classical theory and results of Differential Geometry: The geometry of curves and surfaces lying (mostly) in 3-dimensional space. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is adapted to the curve near that point. Given a curve, one can define two quantities: its curvature and torsion. Both quantities are scalar fields and depend on some parameter which parametrizes the curve that is in general the arc length of the curve. The Fundamental Theorem of Space Curves states that every regular curve in three-dimensional space, with non-zero curvature, is completely determined by its curvature and torsion. It means that from just the curvature and torsion, the vector fields for the tangent, normal, and binormal vectors can be derived using the Frenet–Serret formulas. Then, integrating the tangent field yields the curve. In the first chapter of this thesis, we present the proof of the Fundamental Theorem of Space Curves using two approaches. The first proof is the traditional one used in almost all differential geometry references [1, 2]. The second approach is a new one established recently by H. Guerrrero in [3]. It is based on finding a solution of a nonlinear differential equation of second order. As applications of the second approach, general slants and helices are characterized. The second chapter revolves around defining a parametrized surface in the plane and introducing its first and second fundamental forms. This will allow to define the notions of curvature: the Gaussian curvature and the Mean curvature. The Gaussian curvature describes the intrinsic geometry of the surface, whereas the Mean curvature describes how it bends in space. The Gaussian curvature of a cone is zero: This is why we can make a cone out of a flat piece of paper. The Gaussian curvature of a sphere is strictly positive: This is why planar maps of the earth’s surface invariably distort distances. The Gauss-Codazzi equations (also called the Gauss–Codazzi–Mainardi equations) are fundamental equations which link together the induced scalar product on R3 and the second fundamental form of a surface. The first equation, often called the Gauss equation was discovered by Carl Friedrich Gauss. It states that the Gauss curvature of the surface, at any given point, is encoded by the second fundamental form. The second equation, called the Codazzi equation or Codazzi-Mainardi equation, discovered by Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869) states that the covariant derivative of the second fundamental form is fully symmetric. It turns out that the Gauss-Codazzi equations are sufficient to prove the existence of a surface in R3. This is called the Fundamental Theorem of Surfaces and it is proved in Chapter 3. In fact, consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that satisfy together the Gauss-Codazzi equations in a connected and simply connected open subset of R2. If the matrix fields are respectively of class C2 and C1, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with these fields as its first and second fundamental forms.