MATH3968 – Lecture 38

Exam Study Hints

● Do the tutorial problems.

● Use do Carmo for additional problems

● Check out the past exam papers (2002, 2003, 2005), posted on Canvas.

● All but the first question of the 2005 exam are relevant

● The exam is open book but I suggest you create a short summary (around 2 pages). The real value is in creating it, so no point using someone else’s.

What Types of Questions to Expect?

Questions may include the following:

● Explicit, computational questions (eg 2005: 2, 3; 2003 1,6,7; 2002: 1,4,6,7).

● Applications of major Theorems (eg 2005: 4; 2003: 5; 2002:8)

● Most differential geometry exams have a question on the Gauss-Bonnet theorem or its corollaries

● Questions that require some insight: like a number of your tutorial questions.

● Proofs/examples which slightly modify those from class

● Me not explicitly listing a past question above does not mean it is less relevant.

Rough Summary Of Course

Disclaimer: this list is not exhaustive! The exam may contain things that are not on it. The list of possible questions are by no means exhaustive; they are a place to start, NOT a list of all the things that you need to know.

I’ve picked out some do Carmo (dC) questions for you below for extra practice.

Curves

● arc-length,  curvature, signed curvature  (in plane), torsion,  Frenet frame,  Frenet equations.

● Fundamental Existence and Uniqueness Theorem (proof not examinable)

● rotation index, total curvature, Theorem of Turning Tangents  (proof not exam- inable)

Non-Exhaustive Question Possibilities:

● You could be asked to compute some of the above in an explicit example.

● You could be asked to show that a curve has a particular property.

● you could be asked to work with special types of curves, such as lines of curvature, asymptotic curves, geodesics.

● The Frenet equations are useful.

Example 1 (dC 1.5 - question 4) . Assume that all normals of a regular parametrised curve pass through a fixed point. Prove that the trace of the curve is contained in a circle.

General Analysis

● differential of a smooth map

● Inverse, Implicit Function Theorems (proofs not examinable)

● regular/critical points and values

Non-Exhaustive Question Possibilities:

These are mostly just definitions, although these two theorems are very useful; know their statements.

Example 2 (dC 2.2 - question 7) . Let f (x, y, z) = (x + y + z - 1)2

● a) Locate the critical points and critical values of f

● b) For what values of c is the set f (x, y, z) = c a regular surface?

● c) Answer the questions of parts a) and b) for the function f (x, y, z) = xyz2

Surfaces in R3

● regular surface, surface given as a graph , locally all surfaces given this way, surfaces of revolution, surfaces given as the pre-image of a regular value

● smooth functions, tangent plane, differential of a map

● first fundamental form (metric)

● area, orientation

● Gauss map, second fundamental form, normal curvature, principal curvatures

● Relationship between dN , I and II .

● Gauss and mean curvature

Non-Exhaustive Question Possibilities:

There is so much to compute here, and many possibilities for questions!

● Compute any of the above

● questions involving normal curvature, eg lines of curvature, asymptotic lines, umbilic points

● questions involving curves on surfaces

Example 3 .      1. dC 3.3 - question 2 Determine the asymptotic curves and the lines of curvature of the helicoid, x = v cos u, y = v sin u, z = cu, and show that its mean curvature is zero.

2. dC 3.2 - question 7 Show that if the mean curvature is zero at a nonplanar point, then this point has two orthogonal asymptotic directions.

3. dC 3.2 - question 16 Show that the meridians of a torus are lines of curvature.

● minimal surfaces, fact that they are critical points (in fact local minimiser) for area

● examples of minimal surfaces: catenoid ((only minimal surface of revolution) , he- licoid

Non-Exhaustive Question Possibilities:

Mostly here you should know the theory and the examples.

Example 4 . E.g. dC 3.5 - question 12: Show that there are no compact minimal surfaces in R3 .

● Isometries and local isometries, Minding’s Theorem

● covariant derivatives, Christoffel symbols, Gauss and Codazzi-Mainardi equations (you do not need to memorise these equations, although it is possible that I could ask you to derive them)

● Expression for Christoffel symbols in terms of metric.

● Gauss’s Theorema Egregium

● Fundamental existence and uniqueness theorem for surfaces (Bonnet)

● parallel transport, rotation of vectors under parallel transport

● geodesics,

● algebraic value of covariant derivative, geodesic curvature

● uniqueness and existence theorem for geodesics (proof not examinable)

● exponential map, geodesic polar coordinates, length-minimising property of geodesics

Non-Exhaustive Question Possibilities:

Again, this is core material.

● compute Christoffel symbols

● use in some way the fact that the Gauss curvature is invariant under local isometries

● Give geodesic equations, find geodesics

● compute parallel transport, formula for how angle changes

● geodesic curvature ties in with normal curvature; k2  = kg(2)  + kn(2); use this in some way.

Example 5 .      1. dC 4.2 - question 4 Use the stereographic projection to show that the sphere is locally conformal to a plane

2. dC 4.3 - question 4 Show that no neighbourhood of a point in a sphere may be isometrically mapped into a plane.

Example 6 .      1. dC 4.4 - question 1

● a) Show that if a curve C c S is both a line of curvature and a geodesic, then C is a plane curve.

● b) Show that if a (nonrectilinear) geodesic is a plane curve, then it is a line of curvature.

● c) Give an example of a line of curvature which is a plane curve and not a geodesic.

2. dC 4.4 - question 2 Prove that a curve C c S is both an asymptotic curve and a geodesic if and only if C is a (segment of a) straight line.

Example 7 .      1. dC 4.4 - question 5 Consider the torus of revolution generated by ro- tating the circle

(x - a)2 + z2  = r2 , y = 0,

about the z axis (a  > r  > 0).  The parallels generated by the points (a + r, 0), (a - r, 0),  (a, r) are called the maximum parallel, the minimum parallel, and the upper parallel, respectively. Check which of these parallels is

● a. A geodesic.

● b. An asymptotic curve.

● c. A line of curvature.

2. dC 4.4 - question 6 Compute the geodesic curvature of the upper parallel of the torus of question 5 (above).

The most important theorem we have covered in this course is:

● Gauss-Bonnet theorem (local and global), and its corollaries, including

● Poincare-Hopf

Non-exhaustive possible questions:

● Verify the Gauss-Bonnet theorem on some domain

● Apply Gauss-Bonnet or Poincare-Hopf to prove some global results.

Example 8 (dC 4.5 - question 1) . Let S c R3  be a regular surface homeomorphic to a sphere.  Let Γ c S be a simple closed geodesic in S, and let A and B be the regions of S which have Γ as a common boundary. Let N : S → S2  be the Gauss map of S. Prove that N(A) and N(B) have the same area.