### MA3010: Differential Geometry of Curves and Surfaces

Overview:

The course is devoted to basics on curves and surfaces in the space. The course is split into three parts:

- We start with refreshers on multivariable calculus and linear algrebra which are fundamental tools to the study of curves and surfaces.
- We move on to definition and properties of regular curves in space. Fundamental objects as curvature, torsion and Frenet equations are introduced. We end the analysis of local properties of curves by stating and proving the local canonical form for curves in space. We finally study some global properties of plane curves, in particular the winding number and the rotation index.
- The last part is devoted to the study of regular surfaces. We define regular surfaces as sets with local regular coordinates and we give a wide range of examples: parametrized surfaces, regular values, graphs… Introducing the notion of the tangent plane we study one of the main tool for the study of local properties of surfaces, that is the first fundamental form. Next we consider the question of orientation of surfaces. We end this part and the course by definitions and basic properties of the Gauss map and the second fundamental form.

The course mostly based on the book *Differential Geometry of Curves & Surfaces* by Manfredo Do Carmo, published by Dover.

About the homeworks:

I give two homeworks a month. They consist on two or three exercises using the course and are aimed to help the students to better understant the course, to train themselves on computations and most of all: to make them aware of what they may NOT know or might NOT be able to do. This is perfectly normal and happens all the time, to everyone.

That is why it is important to do them and to hand them – even not completely finished – to me. You can handle them directly to me at the end of a course or put them in my pigeonhole at the School Office.

Please feel free to ask me any question by email at morisseb at cardiff.ac.uk !

Here are the homework sheets:

Schedule:

Courses on Thursday are two hours long. Courses on Tuesday are one hour long, and every other Tuesday. All courses are on Room M/0.40.

- Thursday 04/10

Multivariable calculus (refresher): partial derivatives ; differentiability along a vector ; differential map ; local inverse theorem. - Tuesday 09/10

Linear Algebra (refresher): scalar product and norm in the Euclidean space ; cross product ; basis. - Thursday 11/10

(Linear Algebra: cont’d)

Basics on curves: definition of a regular curve ; change of parametrizations ; parametrization by arclength. - Thursday 18/10

Curves in space: definitions of the curvature and of the torsion. - Tuesday 23/10

Examples - Thursday 25/10

Frenet(-Serret) trihedron and equations. - Thursday 01/11

Local canonical form: statement of the result and its proof.

Summary of the local analysis of regular curves. - Tuesday 06/11

Global properties of plane curves: an introduction. - Thursday 08/11

Global properties (cont’d): the winding number and the rotation index.

Examples regarding the complexity of the global analysis of curves: Jordan theorem ; space filling curve. - Thursday 15/11

Definition of regular surfaces: local coordinates of sets.

Examples: graphs, regular value and parametrized surfaces. - Tuesday 20/11

Examples - Thursday 22/11

The notion of tangent planes: definition through paths on the surface ; link with the image of the differential map of the local coordinates. - Thursday 29/11

The first fundamental form: definition ; computation in local coordinates ; computations of lengths and areas. - Tuesday 4/12

Examples. - Thursday 6/12

Orientation of surfaces: definition and examples.

The second fundamental form: the Gauss map ; definition of the second fundamental form ; basic properties. - Thursday 13/12

Examples.

Summary, Q&A, conclusion.