BSc Mathematics and Philosophy

This unit will be presented as an introduction to Differential Geometry, where the main focus will be on Curves and Surfaces in two- and three-dimensional Euclidean space. As a mathematical discipline, Differential Geometry explores the geometry of smooth shapes. To do this, it uses tools from various areas of mathematics, such as differential calculus, integral calculus and linear algebra.

In Differential Geometry we use analytic and algebraic techniques to formalise the intuitive notions of distance, angle and curvature to study the extrinsic properties of curves (parameterised by arc length). Some of these properties are then used more generally to study the intrinsic and extrinsic properties of surfaces. For instance, since the time of Euclid it was understood that a straight line provides the shortest distance between two points; however, how do you find this distance if the points lie in the surface of the earth (which is not flat). This leads to the important concept of geodesics where one uses the idea that great circles are locally similar to straight lines in a flat plane.

In this module we will explore several notions around smooth parameterised curves, such as tangent vectors, curvature, osculating planes, and the Frenet moving frame. We will then move to the study of parametric surfaces where we discuss tangent planes, parallel transport, curvatures, geodesics, fundamental forms, Christoffel symbols, and the Gauss-Bonnet theorem.  Many examples will be presented throughout.  

The unit aims to deliver a gentle introduction to fundamental aspects of classical Differential Geometry. It explores and studies the local properties of smooth curves and surfaces by means of differential calculus.

• Construct parametric curves in Euclidean space from geometric criteria and define and compute the fundamental notions attached to them such as tangent, normal, and binormal vectors; as well as arc-length, curvature, and torsion.
• State the fundamental theorem of curve theory and global theorems for closed curves, and deduce simple consequences.
• Define and analyze the central notions of Gauss, mean and principal curvatures of surfaces and use them to determine properties of a surface.
• Define and calculate moving frames for curves and surfaces and use them to analyze geometric properties such as Gauss curvature, geodesics and parallel transport.
• Reproduce classical examples of smooth surfaces and compute the first and second fundamental forms in some specific instances; deduce properties of surfaces from these.  
• State and explain the Gauss Theorema Egregium and the Gauss-Bonnet Theorem. Apply to basic examples.

• Parametric Curves (3 weeks). Parametric curves, re-parametrisations, tangent vectors, and arc-length. Scalar curvature, Frenet formulas, and the fundamental theorem of plane curves. Evolute and Envelopes. Examples: circle, ellipse, and envelopes of families of lines.
• Theorems for curves (2 weeks). Total curvature and rotation number. Crofton’s formula. Curvature/Torsion and Frenet formulas. Rigid motions and the fundamental theorem of space curves. Examples: circles and helix. 
• Parametric surfaces (3 weeks). Tangent planes and normal lines (affine and linear). First fundamental form. Curvatures of plane sections and Meusnier’s and Euler’s theorems; Second fundamental form. Principal curvatures.  Gauss and mean curvature. Lines of curvature, parabolic locus and asymptotic lines. Examples to be taken from spheres, surfaces of revolution, ruled surfaces, developable surfaces, and minimal surfaces. 
• Moving frames and applications (3 weeks). Moving frames and associated 1-forms; connection form. Gauss-Codazzi equations and Gauss curvature revisited. Geodesics in terms of tangential curvature. Parallel transport and holonomy. Gauss Theorema Egregium. Gauss-Bonnet Theorem.

Teaching in traditional format with 2hrs of lectures and 1hr tutorial per week.  

Generic feedback made available after exam period

• Differential Geometry of Curves and Surfaces. M. do Carmo, 1976.  
• Differential Geometry: A first course in curves and surfaces. T. Shifrin, 2021.
• An introduction to Differential Geometry. T. J. Willmore, 1964. cs and Parallel Transport (6 lectures, 3 weeks).  Geodesics in terms of curvature. Parallel transport and covariant derivative. Christoffel symbols. Gauss-Peterson-Codazzi equations. Gauss Theorema Egregium. Gauss-Bonnet Theorem.