The differential geometry of curves and surfaces is a subject that aims to use elements from both analysis (including differential and integral calculus) and algebra (including linear algebra and the theory of groups) to study a variety of problems arising in geometry.  Roughly speaking, the differential geometry of curves and surfaces has two aspects.  One is the study of the local properties (i.e. properties which depend only on behaviour in small neighbourhoods) of curves and surfaces; the methods of differential calculus play a large role here.  The second aspect is global in nature, studying the influences of these local properties on the behaviour of the entire curve or surface.

Overall, the purpose of this module is to show students how ideas, particularly from multivariable calculus and linear algebra, can be put together to form an invariant language that is quite useful in understanding the structure of curves and surfaces, which are naturally occurring geometric objects in mathematics.   

• Compute parametrisations of curves and their tangent vectors.

• Comfortably use the Frenet equations and apply their knowledge of linear algebra in this context.

• Understand curvature and torsion of space curves.

• Work with certain theorems about the global properties of plane curves.

• Compute parametrisations of surfaces and their tangent spaces.

• Work with first fundamental forms.

• Understand various properties of the Gauss map.

• Compute the Gaussian curvature of a surface at a point.

27 fifty-minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes during lectures.  Lecture notes will also be posted on Learning Central on weekly basis.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills: How to pass from « local » theory (i.e. calculations from multivariable calculus in n-dimensions) to « global » theory, comfortability with the algebraic structure of various geometric objects, ability to use techniques from linear algebra in settings outside of the algebra course sequence.

Transferable Skills: Ability to understand partial differential equations in the setting of manifolds, algebraic/geometric intuition which can be a base for algebraic topology and algebraic geometry, comfortability in passing between abstract geometric language (topological spaces, groups, forms, etc.) and concrete analytic quantities (charts, arc-length, Gaussian curvature, etc.).

Formative assessment is by means of regular biweekly tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.

The summative assessment is the written examination at the end of the module.  .  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of three from four equally weighted questions.

Tools from Multivariable Calculus:

• Tangent vectors
• Differentials of functions
• Inverse function theorem and implicit function theorem

Tools from Linear Algebra:

• Linearly independent sets

• Spanning sets

• Orthogonality and the Gram-Schmidt Process

Curves:

• Frenet curves and frames
• Curvature and torsion
• Frenet equations and the fundamental theorem of the local theory of curves
• Winding numbers and rotation index
• Global theory of curves: Fenchel's theorem, Hopf's theorem, Fary-Milnor

Surfaces:

• Surface parametrizations
• First fundamental form
• Gauss map and the second fundamental form
• Curvauture of surfaces
• Theorema Egregium

Time permitting:

• Surfaces of rotation
• Rules surfaces
• Minimal surfaces
• Covariant differentiation
• Geodesics
• Gauss-Bonnet Theorem

Kuhnel, W. 2015. Differential geometry; curves, surfaces, manifolds. 3rd ed. American Mathematical Society.

O'Neill, B. 2006. Elementary Differential Geometry. 2nd ed. Academic Press. 1st ed. available as eBook via http://bit.ly/2P9V4Yd

do Carmo, M. 2016. Differential Geometry of Curves and Surfaces. 2nd ed. Dover Publications.