Winter Semester, Academic Year 2019-20, Section B
Contents
Announcements#
Syllabus: Theory of curves in space and in the plane: what is a curve? regular parametrizations and the natural parametrization by length, curvature and torsion, Frenet frame, quadratic approximations, parabola and tangent circle, the Frenet-Serret equations and the fundamental theorem of the theory of curves. Plane curves: sign of curvature, curvature and rate of change of angle.
Elementary theory of surfaces: regular surface parametrizations, basic examples, surfaces of revolution, graphs. First and second fundamental forms, curvature notions, normal curvature, the Gauss map and the shape operator (the Weingarten map.) Maps between surfaces and isometries. Christoffel symbols, compatibility conditions and the Theorema egregium of Gauss. Fundamental theorem of the theory of surfaces (Bonnet).
Schedule: Monday 11-1 (Δ21), Tuesday 12-2 μμ (Δ22) and Thursday 12-1 μμ (Δ21)
Bibliography:
- M. do Carmo Differential Geometry of Curves and Surfaces, 2nd ed. Dover 2016
- A. Pressley Elementary Differential Geometry, 2nd ed. Springer 2010
- M. Lipschultz Schaum’s Outline of Differential Geometry, McGraw 1969
- Αρβανιτογεώργος Α. Στοιχειώδης Διαφορική Γεωμετρία, Κάλλιπος 2015 (free access)
Study Guide: As with every subject in Mathematics, the differential geometry of curves and surfaces requires regular involvement with the course material in order to be absorbed properly. Most will benefit from a good set of class notes. Regular revision (every week at least) is necessary, so that all concepts presented are well understood. But this is not sufficient! it has to be complemented by a more active study, practising first of all all the techniques given in class using examples from books but also made up by the student, followed by attempting more challenging material. It is of course critical that you do not fall behind. Make it your aim to “conquer” this new subject, by building your own ways of interpreting and comprehending the material. Do not be afraid of mistakes and do not hesitate to seek help from the lecturer, as well as from your fellow students.
And finally, some similar advice from colleagues here and here.
Calendar#
[For Erasmus students not attending classes, this calendar gives information about coverage of material to assist your study. The assignments are posted in English. Helpful visual material is also posted.]
| Class/Hours | Material covered |
Files, Links |
Assignments |
| 30-9-2019 [2,2] | Introduction to CDG: notions of curvature of a curve, need for a second notion, the torsion, for space curves, examples. What is a surface, derivative and linear approximations. | ||
| 1-10-2019 [2,4] | Curvature of surfaces, local types (elliptic, hyperbolic, parabolic.). Review: vector spaces, bases, inner products. | ||
| 3-10-2018 [1,5] | Inner products and positive definite matrices, examples. | ||
| 7-10-2019 [2,7] | Criterion for a 2Χ2 matrix to be +ve definite, orthonormal bases and ease of finding coordinates, projection to a subspace, Gram-Schmidt process, geometric interpretation and examples. The exterior product in space. | ||
| 8-10-2019 [2,9] | Calculation of exterior product, triple product and their geometric meaning. The derivative of a function at a point as a linear map. Inverse function theorem, changes of variables and examples. | ||
| 10-10-2019 [1,10] | Implicit function theorem, geometric content and examples. Curves, convenient way to define them. | ||
| 14-10-2019 [2,12] | Regular curve parametrizations and examples. Taylor expansions, linear approximation of a curve at a point, notions of acceleration, in preparation for defining curvature. | ||
| 15-10-2019 [2,14] | Tangent and normal components of acceleration, if speed is constant acceleration is orthogonal to motion. reparametrizations, examples, natural length parametrization. | ||
| 17-10-2019 [1,15] | Definition of curvature function κ(s) and unit normal n(s), the orthonormal moving Frenet frame. | ||
| 21-10-2019 [2,17] | Contact of order k between curves, application: tangent line (1st order contact) and parabola (2nd order contact). Tangent circle, proof that it also has 2nd order contact. | Tangent plane to helix at a point | Assignment 1 |
| 22-10-2019 [2,19] | Frenet frame, Frenet-Serret equations and definition of torsion. Computation of torsion for the helix. | ||
| 29-10-2019 [2,21] | Types of curves as solutions of the F-S equations. Computation of the torsion (formula). Finding the Frenet frame/trihedron at a point from non-natural parametrisation. | Frenet frame for helix (YouTube video by V. Papageorgiou) | |
| 31-10-2019 [1,22] | Two methods for finding the Frenet frame at a point and computation of the curvature function, both from arbitrary parametrisation. | ||
| 4-11-2019 [2,24] | Computation of torsion from arbitrary parametrisation. Planar curves: defining first the orthonormal basis (t(s), n(s)), then the curvature κ(s), which now can take negative values. | ||
| 5-11-2019 [2,26] | Tangent circle and parabola at a point. Difficulties in defining “angle”, definition of total change of angle as an integral | ||
| 7-11-2019 [1,27] | Curvature as rate of change of angle. | ||
| 11-11-2019 [2,29] | Theory of surfaces: regularity condition explained, regular surface parametrizations. Examples: planes, cylinders, spheres | ||
| 12-11-2019 [2,31] | Tangent vectors and tangent plane to a point of a surface. Unit normal vectors, examples. | ||
| 14-11-2019 | No class held since building was under occupation. | ||
| 18-11-2019 [2,33] | The area element du.dv goes to the area element  , the sphere as an example. Graphs are surfaces. Orientability of surfaces. |
||
| 19-11-2019 [2,35] | Two orientations exist for connected orientable surface. Non-orientable surfaces (e.g. Möbius strip). Regular surface parametrisation always gives orientation. Surfaces of revolution: orthogonal transformations, rotations. Regularity conditions for SoR. | ||
| 21-11-2019 [1,36] | Tutorial: worked on problems from homework and resolved questions | ||
| 25-11-2019 [2,38] | Examples of SoR: cylinders, spheres, tori. Computing the area on a surface, examples. Definition of a surface. | ||
| 26-11-2019 [2,40] | Change of parametrization, example. Measurements on a surface, need to work through parametrization. The scalar product transforms to an inner product on the parameter plane. | ||
| 27-11-2019 [2,42] | The First Fundamental Form (FFF) for a parametrisation, examples. Curves on a surface and their length, examples, loxodrome. | ||
| 28-11-2019 [1,43] | Tutorial: worked on problems from homework and resolved questions | ||
| 2-12-2019 [2,45] | The loxodrome, continued. Computing the area on a surface using the FFF. Curvature of a surface: first approach: deviation of nearby points from the tangent plane at a point, definition of the Second Fundamental Form (SFF). | Loxodrome | |
| 3-12-2019 [2,47] | The SFF for graphs, relation with the Hessian matrix. Classification of points of a surface (elliptic, hyperbolic, parabolic.) | ||
| 4-12-2019 [1,48] | 2nd approach: variation of the tangent planes, the Gauss map. Image of the Gauss map, examples. Aim: find its derivative. | ||
| 9-12-2019 [2,50] | Geometric interpretation of the derivative of function  . Definition of a smooth map between surfaces, the Gauss map as an example.Computation of the derivative of the Gauss map through the matrices of the FFF and the SFF. |
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| 10-12-2019 [2,52] | SFF zero <=> surface is planar. 3rd approach: using curves on the surface through a point. Normal component of the acceleration and definition of the shape operator. The normal curvature. | ||
| 11-12-2019 [1,53] | Proof of the symmetry of the shape operator. Principal curvatures and directions and Gauss and mean curvature. How are these to be computed? | ||
| 12-12-2019 [1,54] | Tutorial: worked on problems from homework and resolved questions | ||
| 16-12-2019 [1,55] | Overview of the 3 approaches to curvature of surfaces. Computing principal curvatures, example of cylinder. The Gauss curvature as the ratio of the determinants of the second and first FF. | ||
| 17-12-2019 [2,57] | Computational examples, contd: spheres, graphs. The Euler formula for the normal curvature function. | ||
| 18-12-2019 [1,58] | Classification of points based on Gauss and mean curvatures K and H, asymptotic directions. Examples: saddle, the monkey saddle, torus. | ||
| 19-12-2019 [1,59] | Tutorial: problems from homework and questions from class | ||
| 23-12-2019 [2,61] | Change of parametrisation and invariance of Gauss curvature. Defining the derivative of a map between surfaces. Local isometries between surfaces, examples. | ||
| 9-1-2020 [1,62] | Local isometry examples: cone and plane, catenoid and helicoid surfaces. | Visualising isometry between catenoid and helicoid | |
| 13-1-2020 [2,64] | PDEs for the moving frame  on a surface, Christoffer symbols: they only depend on the elements of the FFF and their 1st-order derivatives. |
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| 14-1-2020 [2,66] | Compatibility conditions, equations of Codazzi-Mainardi and of Gauss. The theorema egregium of Gauss. Isometries preserve the Gaussian curvature, examples. END |