BSc Actuarial Science and Mathematics

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

• Parametric Curves (6 lectures, 3 weeks). Velocity, tangent vectors and arc length. Acceleration and curvature. Torsion of space curves. Independence of parameterisation, osculating plane/circle/sphere and idea of n-point contact. Frenet moving frame and Serret-Frenet formula. Evolute and Involute. Examples: conics, helix, and envelopes of families of lines.
• Theorems for plane curves (4 lectures, 2 weeks).  Rigid motions and the fundamental theorem of curve theory. Crofton’s formula. Total curvature and rotation number.
• Parametric surfaces (6 lectures, 3 weeks).  Tangent planes (affine and linear). First fundamental form. Curvatures of normal sections. Euler’s theorem and principal curvatures. Gauss and mean curvature. Second fundamental form. Meusnier’s theorem. Lines of curvature, parabolic locus and asymptotic lines. Examples to include spheres, ruled surfaces, developable surfaces, and minimal surfaces.  
• Geodesics and Parallel Transport (6 lectures, 3 weeks).  Geodesics in terms of curvature. Parallel transport and covariant derivative. Christoffel s

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.