## Lectures and Exercises will be held online by means of interactive video sessions. You can join using this Zoom link on Monday from 13h15 to 17h.

#### Horaires:

Cours: lundi de 13h15 à 15h, salle MAA331
Exercices: lundi de 15h15 à 17h, salle MAA331

## Cours

#### Requirements

1st and 2nd year courses in math or physics, (or with teacher's permission)

##### Recommended

Ordinary Differential Equations, BA Math (MATH-301)

#### Contents

This course will describe the classic differential geometry of curves, tubes and ribbons, and associated coordinate systems. We will prove various classic mathematical theorems such as the Weyl-Hotelling formula for tube volumes, and the relation between Link, Twist and Writhe, which couples differential geometry and topological invariance for closed and knotted framed curves. While we will not consider applications explicitly in this course, much of the mathematical material that will be described is central in various problems of mechanics, including nanostructures and topological fluid mechanics.

##### Parts:

1) Framed Curves--basic differential geometry of curves in the group SE(3) of rigid body displacements

2) The Calugareanu-Fuller-White Theorem Lk = Tw + Wr. Topology meets vector analysis and non-uniform convergence

3) The geometry of Coordinates on SO(3) and 2pi vs 4pi. Euler angles, Cayley vectors, Euler parameters, and quaternions.

4) Fattened curves, Tubes and Ribbons. Contact framings, global radius of curvature, and ideal shapes.

#### Lecture notes

These notes are meant to supplement your personal notes. They are to be understood as a first draft and as such may contain inaccuracies and mistakes. Be critical and do not hesitate to let us know if you find errors and/or typos. Furthermore, they make no pretence of being exhaustive. The material of the course is by definition what is exposed during lectures and exercise sessions. Finally, as the semester progresses, come back to the website and check frequently: pay attention to the version numbers. The notes will be edited as we progress during semester. The original document is a collection of chapters corresponding to the different lectures given last year. The horizontal red line indicates where the oral lecture is at.

The keen student, will also find relevant material on last year's webpage.

There is an older polycopie associated with a DNA modelling masters course with some chapters, specifically chapters 8 and 9 on this page. This material will be incorporated in the new polycopie in due course.

#### Week-by-week correspondence

 Week 1 (14.09) Overview of course and physical demonstrations of the Calugareanu $Lk=Tw+ Wr$ formula. Basic vector and matrix notation. Arc-length, curvature, (geometrical) torsion, and Serret-Frenet equations for a space curve. The notes of the first lesson are here. You can download the recorded lesson by means of video 1 and video 2. (Last year notes of the first lesson are here). Week 2 (28.09) The Lie groups $O(3)$, $SO(3)$, and $SE(3)$. Left and right actions in $SO(3)$ and $SE(3)$, both algebraic and geometric interpretation. Curves in $SO(3)$ and in $SE(3)$. The notes of the second lesson are here. You can download the recorded lesson by means of the following video. (Last year notes of the second lesson are here). Week 3 (05.10) Darboux vector of a curve in $SE(3)$. Darboux vector of the Frenet frame. Framed curves, intrinsic, extrinsic, adapted or not. Factorisations of curves in $SO(3)$ and relations between their Darboux vectors. Relations between two adapted framings particularly important. Factored curves in $SE(3)$, and the special case of two offset curves in $\mathbb R^3$. The notes of the third lesson are here. You can download the recorded lesson by means of the following video. (Last year notes of the third lesson are here). Week 4 (12.10) Statement of Calugareanu Theorem for a smoothly closed curve $x$ and a smoothly closed offset curve $y$. First definitions of a) Link $Lk$ of two closed non-intersecting curves, b) Total twist $Tw$ of a unit normal field about a curve $x$, and c) Writhe $Wr$ of a non-self-intersecting curve $x$. Start of discussion of the properties of Link. The notes of the 4th lesson are here. You can download the recorded lesson by means of the following video. (Last year notes of the 4th lesson are here). Week 5 (19.10) Further properties of Link. Computing Link by homotopy to a sum of Hopf links. Rules for computing Link via a count of signed crossings in one particular projection. Start of relating signed crossing formulas for Link to the double integral definition via the signed area formulas on the unit sphere. The notes of the 5th lesson are here. You can download the recorded lesson by means of the following video. (Last year notes of the 5th lesson are here). Week 6 (26.10) End of discussion of the boundary of the zodiacus, and singularities of the projection of the surface $y(\sigma) - x(s)$ (compare with last weeks exercises). Connection between signed area integral definition of Link and counts of signed crossings in one specific (generic) projection. The video together with the Mathematica files and images used in the demonstration of Link as signed area are in this archive. The notes of the 6th lesson are here. You can download the recorded lesson by means of the following video. (Last year notes of the 6th lesson are here). ----------Updated till here ---------- Week 7 (30.10) Curves lying on general surfaces, and the surface normal extrinsic adapted framing. The special case of curves lying on the unit sphere, and the particular case of the tangent indicatrix or tantrix of curve. Introduction to the next exercise series, tantrices of closed curves. Discussion of properties of Writhe. Interpretation as signed area of the zodiacus (now of one curve instead of two as was the case for Link), and singularities/discontinuities in the boundary of the zodiacus coming from the tantrix of the curve x. Interpretation of writhe in terms of global radius of curvature circles, namely as the sign-indefinite weighted sum of radii of circles that pass through two points of the curve and are tangent at one point. The notes of the seventh lesson are here. Week 8 (06.11) Proof of the main C-F-W Theorem. Read through Chapter 7 of the polycopie. Note that Chapter 7 was originally written as a standalone document, so much of the first part of the Chapter is a rapid review of material that has already been treated in more detail in the current version of the polycopie. For an expanded version of the proof you can read the hand-written notes of Giulio Corazza here. Week 9 (13.11) Strand passages and curves with an even or odd 'number of turns'. Register angle between two framings. Discussion of particular framings of curves: Frenet-Serret, surface, and parallel transport framings. The notes of the ninth lesson are here. Week 10 (20.11) Continuation of special (throughout always assumed to be adapted) framings of closed curves: completion of natural framings, and introduction of the writhe framing (which is always closed and zero link for any smoothly closed curve). Interpretation of $Tw + Wr$ as the discontinuity angle for closed curves with open framings, and the lemma that framings of closed curves are closed iff $Tw + Wr$ is an integer. Open problem (not examinable): for framed open curves with a spherical closure between the point-tangent data pairs at each end (and with surface normal framing, and either for general curve closure or with biarcs) is there a simple geometrical interpretation of the sum $Tw + Wr$? The notes of the tenth lesson are here. Week 11 (27.11) The midpoint rule applied to first order linear matrix ODE to motivate the Cayley transform. In the case of the matrix group $SO(3)$ connections between the Darboux vector and the Cayley vector of the Cayley transform (sometimes also called the Gibbs vector). Properties of the Cayley transform and the Euler-Rodrigues formula in terms of the Cayley vector. The notes of the eleventh lesson are here together with the corrigendum. Week 12 (04.12) Euler-Rodrigues Parameters as a double covering of $SO(3)$ to avoid singularities. Euler-Rodrigues formula in terms of the Cayley vector and connections to the Euler-Rodrigues parameters via stereographic projection. Discussion of the multiply covered circle in light of Euler parameters and tracking rotations mod $4\pi$ along curves of rotation matrices. Besides the polycopie (part III, chapter 9), see also these notes from Dichmann. The notes of the twelfth lesson are here. Week 13 (11.12) Geometrical interpretation of a rotation as product of two reflections. Householder reflection matrices. Composition rule for Euler-Rodrigues parameters based on geometrical reflection decomposition. You can find the proof of the latter statement in Dichmann's notes. Composition rule for Cayley vectors exploiting the one for Euler parameters. It is also possible to recover the same composition rule starting from the definition of Cayley transform but the proof is somehow tedious and long. Introduction to quaternions. Who wants to give a look to the geometrical proof of the composition rule directly derived for Cayley vectors can read these notes by Thomas Zwahlen. Moreover, you may find more information about the parametrization of the rotation group in the article section of the page, where the 1840 original paper by Rodrigues is included. The notes of the thirteenth lesson are here. Week 14 (18.12) Volume of a tube. Condition for local self-intersection avoidance. Equilibrium of strings and the case of frictionless contact in particular. Here are notes meant to complement your own. Moreover, you can find the images showed in class. The notes of the fourteenth lesson are here.

#### Summary and description of the exercices

This document contains an overview and a description of all the exercises given so far.

## Bibliography

There is no text book that we are aware of covering the material of this course. The first part on Frenet frames is however very standard and is discussed in any book on the Elementary Differential Geometry of Curves and Surfaces, of which there are many. One good one is by D. J. Struik, and another (from which some of the series questions were taken) is by M. P. Do Carmo.

The citations below are to research or survey articles concerning the material of the course. The citations have links to PDF versions of the articles but for copyright reasons the links are restricted to students in the class via password protection, login: frames, password as given in class.

Two articles, partially contradictory, regarding the history of the Gauss linking number :

Three survey articles, some pages from a standard undergraduate text book, and an introductory monograph all discussing link, twist, and writhe, and the Calugareanu-Fuller-White (CFW) theorem including connections with DNA

An article on magnetohydrodynamics which relates helicity of a vector field to the CFW. The introduction contains a historical perspective of the CFW theorem.

An article on scroll waves, with an extensive discussion of the CFW theorem, including another historical perspective.

A topology textbook which gives a friendly introduction to the subject. Chapters 10 and 14 are the most pertinent to the course.

A beautiful little book "A singular mathematical promenade" by Étienne Ghys. The most pertinent chapter for this course is the second to last one all about the Gauss linking number.

Some articles on the parametrization of the 3-dimensional rotation group. In particular, the second one talks about the history of the composition rule and you can also find an explanation of the formula derived by Rodrigues in its original paper (third of the list).

#### More technical articles

The three original articles by Călugăreanu (the third one in Romanian):
Three original articles by Fuller:
An article that gives much more detailed proofs of what Fuller states plus original results on how Writhe changes under homotopy
An article that outlines a geometric point of view of the proof of CFW: the proof comes as a direct application of Stokes theorem on the pull-back of a certain 2-form. Note however that a discussion of the smoothness of the curves concerned is missing as well as a discussion of why it is that the various integrals defined do converge. A similar, and perhaps more accessible, discussion of Stokes Theorem applied to the spanning Seifert surface of a closed curve (including knotted curves) are also presented in the Tyson and Strogatz article cited above.
Articles on the differential geometry of closed curves :
Ribbons in other application areas