MATH3968 – Lecture 38
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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.
Abstract Manifolds
● abstract manifolds, and all the relevant definitions: eg smooth functions, tangent vectors, differential, orientation
● Riemannian metric, covariant derivative, Christoffel symbols, Levi-Civita covariant derivative, geodesics.
● tangent bundle
● Examples: RPn , R2 /Z2 , Klein bottle, hyperbolic plane, tangent bundle.
● Gauss-Bonnet on an oriented abstract surface with Riemannian metric.
Non-exhaustive possible questions:
● Check that you know definitions, and that you understand why they make sense.
● Examples: prove that something is a manifold, or is/is not orientable.
● Explicitly compute some of the above, such as covariant derivatives, geodesics.
● Apply Gauss-Bonnet to an abstract orientable surface with Riemannian metric, such as the hyperbolic plane or the flat torus.
Example 9 (dC 5.10 - question 1) . Introduce a metric on the projective plane RP2 so that the natural projection π : S2 → RP2 is a local isometry. What is the (Gaussian) curvature of such a metic?
Unit of Study Survey:
https://student-surveys.sydney.edu.au
2021-11-30