# Math 140: Differential Geometry

## Instructor

Michael Hutchings
hutching@math.berkeley.edu.
Office: 923 Evans.
Tentative office hours: Thursday 9:00-11:00

## Piazza

You can ask questions and have discussions at the piazza site here.

## Textbook

• R. Millman and G. Parker, Elements of Differential Geometry. This is the required texbook for the course. All references below are to this book unless otherwise stated.
• M. do Carmo, Differential Geometry of Curves and Surfaces, revised and updated second edition. This book is not required, but recommended for supplementary reading; it goes into more depth on some of the topics of the course.

## Syllabus

This course is about the geometry of curves and surfaces in three-dimensional space. We will also study the "intrinsic" geometry of surfaces: that is, geometric notions which are described just in terms of the surface and not in terms of an embedding of the surface into three- or higher-dimensional space. A central theme is to study different kinds of curvature - defined locally on a curve (in chapter 2 of the book) or surface (in chapter 4) - and how curvature relates to global properties of the curve or surface (in chapters 3, 5, and 6). One of the main results in this direction which we will prove near the end of the course is the Gauss-Bonnet theorem, and we will also see several others. If time permits we will give an introduction on how to generalize differential geometry from curves and surfaces to spaces (manifolds) of any dimension (chapter 7).

• Curved homework score: 15%
• Curved midterm score: 35%
• Curved final exam score: 65%
• Minimum of curved midterm score and curved final exam score: -15%

## Homework

Homework assignments will be due on Tuesdays at the beginning of class. Note: To get full credit, please write clear solutions using complete English sentences (with mathematical symbols inserted as appropriate). You are welcome to collaborate or look things up, but you must write your own solutions and cite any sources or collaborators.
• HW#1, due 2/5: Section 2.1 problem 7; Section 2.2 problem 5; Section 2.3 problems 2, 7; Section 2.4 problems 1, 4, 6, 14, 18; Section 2.5 problem 2.
• HW#2, due 2/12: Section 2.5 problems 3, 5, 7; Section 2.6 problems 7, 8; Section 3.2 problems 2,3.
• HW#3, due 2/19: section 4.1 problems 1, 2(a), 4, 5, 9, 10, 11, 14. (Mostly computations this time, but please always write sentences to explain what you are doing.)
• HW#4, due 2/28: section 4.3, problems 1 and 2; section 4.4, problems 2, 4, 5, 9; section 4.5, problems 2 and 5. (Because I am nice, I am not assigning section 4.4, problem 11. You're welcome.)
• HW#5, due 3/14 (pi day!): section 4.5, problems 6, 9; section 4.6, problems 2, 5; section 4.7, problems 1, 5, 6, 7.
• HW#6, due 3/21: section 4.8, problems 8, 15, 16, 37, 39 (for 39, curvature means Gauss curvature); section 4.9, problems 9.5, 9.8 (first sentence).
• HW#7, due 4/9: section 4.9, problems 6, 9; section 4.10, problems 5, 6, 7, 8; section 4.11, problem 8.
• HW#8, due 4/18: section 5.2, problems 1, 2, 3; section 6.1, problem 2; section 6.2, problems, 1, 2, 3. For the last problem you can use the fact that K = <R(X,Y)Y,X> when X and Y are orthonormal.
• HW#9, due 4/30: Section 6.4, problems 1,2; section 6.5, problems 1, 2, 3, 5. (For the last problem, you need to assume that the surface is connected, and "looks like" means "is homeomorphic to".)

## Lecture summaries and references

After each lecture, summaries and reading suggestions will be posted here.
• (Before the course) You might want to review Chapter 1 (Preliminaries) of the book. This is background material which you are supposed to have seen before (there are additional prerequisites which will come up later), and I am not planning to spend time on it in class.
• (Tuesday 1/22)
• Introduction to differential geometry.
• Regular curves, tangent vectors, arc length. See sections 2.1 and 2.2 of the book. Next time we will continue to discuss the material in chapter 2.
• (Thursday 1/24)
• How to reparametrize a curve by arc length. See section 2.2.
• Curvature and torsion. See section 2.3. Warning: do Carmo uses the opposite sign convention for torsion.
• Frenet-Serret formulas. See the beginning of section 2.4.
• (Tuesday 1/29)
• Applications of the Frenet-Serret formulas. See chapter 2.4
• Statement of the existence and uniqueness of curves with prescribed curvature and torsion. See chapter 2.5.
• A bit extra about how to prove the fundamental existence and uniqueness theorem for solutions of ODEs (stated without proof as Thm. 2.5.1 in the book). For full details see section 1.2 of this nice book by Taylor (legal free download from the UC Berkeley computer network). This is an important result with a beautiful proof which everyone should be familiar with.
• (Thursday 1/31)
• Finished the proof of the existence and uniqueness of curves with prescribed curvature and torsion. This is in section 2.5. (I am skipping over section 2.6; this contains some calculations which you can refer back to if needed.)
• Started discussing the "rotation index" (I would call it "rotation number") of plane curves. See section 3.2. Also, please make sure you are comfortable with the background material in section 3.1 on line integrals and Green's theorem, as I am not planning to review this in class.
• (Tuesday 2/5)
• More about the rotation number of a plane curve. See section 3.2.
• (Thursday 2/7)
• Convex curves. See section 3.3.
• The isoperimetric theorem in the plane. See section 3.4. I did not explain the proof in the book, but instead sketched a different proof. For more about this and some other proofs, see e.g. the survey "The isoperimetric problem on surfaces" by Howards-Hutchings-Morgan. (I'm not allowed to post it here but you can access the article on jstor from the UC Berkeley computer network).
• (Tuesday 2/12)
• Review of derivatives of multivalued functions, the chain rule, and the inverse function theorem. For more about this see sections 1.1 and 1.3 of the Taylor book linked above (1/29).
• "Simple surfaces", tangent vectors. See section 4.1 of the book.
• (Thursday 2/14)
• Definition of a surface. See section 4.2.
• Tangent space. See sections 4.1 and 4.2.
• Implicit function theorem. (For the proof of the hard part of this, see the Taylor book above.)
• (Tuesday 2/19)
• The metric (first fundamental form) on a surface. Upper and lower indices and Einstein summation convention. See Section 4.3.
• Geodesic curvature of a curve on a surface. See Section 4.4.
• Started on the second fundamental form and Christoffel symbols. See Section 4.4. We will explain much more about this next time. Our goal, which will take some time, is to understand curvature of a surface.
• (Thursday 2/21)
• Christoffel symbols and indices, oh my! See section 4.4.
• Started discussing geodesics. See section 4.5.
• (Tuesday 2/26)
• Example: geodesics on a surface of revolution. (See section 4.5)
• Proof that length minimizing curves are geodesics. (See section 4.5.)
• (Thursday 2/28)
• Covariant derivative and parallel transport. (Parallel transport is discussed in section 4.6 of the book; the covariant derivative is only discussed in chapter 7 in a more general context.)
• (TUESDAY 3/5) MIDTERM IN CLASS (note date change!). You may bring one formula sheet to the exam. No other aids will be allowed. The exam covers material from lectures, readings, and homework up to section 4.5 (geodesics).
• (Thursday 3/7)
• More about covariant derivative and parallel transport. See section 4.6.
• Started discussing the second fundamental form and the Weingarten map (which are essentially the same thing, for reasons to be explained next time). See section 4.7. Next week we will use this to discuss the curvature of a surface!
• (Tuesday 3/12)
• How the second fundamental form and the Weingarten map are equivalent via raising or lowering an index. See section 4.7.
• Principal curvatures. See section 4.8.
• (Thursday 3/14)
• Digression on tangent vectors as derivations.
• Definition of mean curvature and Gauss curvature.
• Area of a surface.
• Relation between mean curvature and variation of area. (I just stated this; it is not in the book.) Started talking about minimal surfaces. See section 4.8
• (Tuesday 3/19)
• Clarification of why the definition of area does not depend on the choice of coordinate patch.
• Definition of a smooth map between surfaces and the derivative of a smooth map. The former is defined in section 4.10; the latter is unfortunately not defined until section 7.5, in a more general context; you can try to read this but it might not be comprehensible yet.
• Example: the derivative of the Gauss map is minus the Weingarten map.
• More remarks about the geometric significance of mean curvature and Gauss curvature.
• (Thursday 3/21)
• Clarification about the geometric meaning of Lie bracket. (This isn't really in the book.)
• Isometries. See section 4.10. (Compared to the book, I explained more of the theory, but didn't get to all the examples.)
• (Tuesday 4/2)
• More examples of isometries; see section 4.10.
• A bit about surfaces of revolution with constant (Gauss) curvature; see section 4.11.
• A bit more about the more abstract concepts in section 4.9.
• (Thursday 4/4)
• And a bit more about the abstract material in section 4.9.
• Fenchel's theorem; see section 5.1.
• (Tuesday 4/9)
• Crofton's formula and the Fary-Milnor theorem. See section 5.2. (The book's discussion of Crofton's formula is a bit lacking in rigor when it talks about area.)
• Started discussing global properties of surfaces from section 6.1.
• (Thursday 4/11)
• More about global properties of surfaces from section 6.1.
• Geodesic coordinate patches; see section 6.2.
• (Tuesday 4/16)
• Proved that two surfaces with the same constant Gauss curvature are locally isometric. See section 6.2.
• (Section 6.3 is trying to explain some topology, but it is a bit confusing and contains at least one incorrect statement, so I wouldn't read it too carefully.)
• Proved most of the Gauss-Bonnet formula for a simply connected region. See section 6.4.
• (Thursday 4/18)
• More about the Gauss-Bonnet formula; see section 6.4.
• The Gauss-Bonnet theorem and Euler characteristic; see section 6.5.
• (Tuesday 4/23)
• Proof that a compact surface with positive Gauss curvature is convex; see section 6.6. (We explained some more of the topology in class,)
• (Thursday 4/25)
• Discussion of topological manifolds, especially the two-dimensional case.
• Definition of a smooth manifold. See section 7.2. (Note: the book assumes that manifolds are metric spaces. This is nonstandard. The usual thing to do is to assume that the underlying topological space is second countable and Hausdorff. The metric space hypothesis implies these properties, but we don't want to fix a distance function in advance, since later a Riemannian metric will give us one.)
• (Tuesday 4/30)
• More about smooth manifolds. Smooth functions. The tangent space. See sections 7.2 and 7.3.
• (Thursday 5/2)
• More synthesis of the course material from the more general perspective of chapter 7.
• (Tuesday 5/7) (RRR week, review session)
• (Thursday 5/9) (RRR week, review session)
• (FRIDAY 5/17) FINAL EXAM, 7-10PM (in the usual classroom)